Stefan problems for reflected SPDEs driven by space–time white noise

SPA: 3494

Model 1

pp. 1–37 (col. fig: NIL)

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Stefan problems for reflected SPDEs driven by space–time white noise Ben Hambly, Jasdeep Kalsi ∗ Mathematical Institute, University of Oxford, United Kingdom Received 19 August 2018; received in revised form 22 February 2019; accepted 1 April 2019 Available online xxxx

Abstract We prove the existence and uniqueness of solutions to a one-dimensional Stefan Problem for reflected SPDEs which are driven by space–time white noise. The solutions are shown to exist until almost surely positive blow-up times. Such equations can model the evolution of phases driven by competition at an interface, with the dynamics of the shared boundary depending on the derivatives of two competing profiles at this point. The novel features here are the presence of space–time white noise; the reflection measures, which maintain positivity for the competing profiles; and a sufficient condition to make sense of the Stefan condition at the boundary. We illustrate the behaviour of the solution numerically to show that this sufficient condition is close to necessary. c 2019 Elsevier B.V. All rights reserved. ⃝

1. Introduction Stefan problems have been extensively studied since the original work by Josef Stefan in 1888, and have a number of applications in physics, engineering, biology and finance. Broadly speaking, they describe situations where an interface moves with pressure or relative pressure due to competition from two types. In this paper, we will study stochastic, reflected versions of this problem in one-dimension, which take the form

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∂u 1

= ∆u 1 + f 1 ( p(t) − x, t, u 1 (t, p(t) − ·)) + σ1 ( p(t) − x, t, u 1 (t, p(t) − ·))W˙ + η1 ∂t (1.1) ∂u 2 2 2 2 2 ˙ = ∆u + f 2 (x − p(t), t, u (t, p(t) + ·)) + σ2 (x − p(t), t, u (t, p(t) + ·))W + η , ∂t ∗ Corresponding author.

E-mail addresses: [email protected] (B. Hambly), [email protected] (J. Kalsi). https://doi.org/10.1016/j.spa.2019.04.003 c 2019 Elsevier B.V. All rights reserved. 0304-4149/⃝

Please cite this article as: B. Hambly and J. Kalsi, Stefan problems for reflected SPDEs driven by space–time white noise, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.04.003.

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where, at any time t ≥ 0, u 1 is supported on [ p(t) − 1, p(t)], u 2 is supported on [ p(t), p(t) + 1] and u 1 (t, p(t) − 1) = u 1 (t, p(t)) = u 2 (t, p(t)) = u 2 (t, p(t) + 1) = 0. The point p(t) evolves according to the equation ) ( 1 ∂u 2 ∂u (t, p(t)− ), (t, p(t)+ ) . p ′ (t) = h ∂x ∂x ˙ Here, W is a space–time white noise; f i and σi , for i = 1, 2, are the drift and volatility for each type, which are determined by the distance to the moving boundary p(t); and (η1 , η2 ) are reflection measures for the functions u 1 and u 2 respectively, keeping the profiles positive and satisfying the conditions ∫∞∫ (i) 0 R u 1 (t, x) η1 (dt, dx) = 0, and ∫∞∫ (ii) 0 R u 2 (t, x) η2 (dt, dx) = 0. The equation describes the evolution of two reflected SPDEs which share a moving boundary. The derivative of the moving boundary is then determined by h, a locally Lipschitz function of the spatial derivatives of the two SPDEs at the shared boundary. We note here that, in general, reflected SPDEs of this type will only be up to 1/2-H¨older continuous in space. We will therefore require the functions σi to satisfy suitable conditions which will ensure that the volatility decays at least linearly at the interface, which will be shown to be sufficient for the existence of a spatial derivative there. We recall the classical one-sided version of the Stefan problem, which takes the form ∂u ∂u = ∆u, p ′ (t) = − (t, p(t)+ ). ∂t ∂x The profile of u at a given time t has support in the set [ p(t), ∞), and we have the Dirichlet condition u(t, p(t)) = 0. Typically, we have that u 0 ≥ 0, from which it follows that u ≥ 0. The equation for p ′ then implies that the boundary recedes with “pressure” from u. This can be thought of as a model for the melting of a block of ice when in contact with a body of water. The point p(t) represents the interface between the water and ice, and u the temperature profile of the body of water. Recently, stochastic perturbations of this classical problem have received significant attention. In [6], existence and uniqueness for solutions to a Stefan problem where the two sides satisfy SPDEs driven by spatially coloured noise is proved. A particular case of the corresponding problem when the SPDE is driven by space–time white noise was then studied in [10]. It is assumed in [10] that the volatility, σ (x) vanishes faster than x 3/2 as x ↓ 0 i.e. as the moving interface is approached. More recent work on such problems include the models in [7] and in [5]. In these papers, as in [6], the two sides satisfy SPDEs driven by spatially coloured noise, as this makes it easier to establish the existence of the spatial derivative at the interface. Motivated by modelling the limit order book in financial markets, the authors also include a Brownian noise term in the moving boundary. In addition to stochastic moving boundary problems, the study of reflected SPDEs has also attracted interest. Equations of the type ∂u = ∆u + f (x, u) + σ (x, u)W˙ + η, ∂t where W˙ is space–time white noise and η is a reflection measure, were initially studied in [8] in the case of constant volatility i.e. σ ≡ 1. Existence for the equation in the case when σ = σ (x, u) satisfies Lipschitz and linear growth conditions in its second argument was then Please cite this article as: B. Hambly and J. Kalsi, Stefan problems for reflected SPDEs driven by space–time white noise, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.04.003.

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proved in [3] using a penalisation method. Uniqueness for varying volatility σ = σ (x, u) on compact spatial domains was then shown in [9], with the authors achieving this by decoupling the obstacle and SPDE components of the problem. In this paper, we aim to take a first step towards studying reflected stochastic Stefan problems, by proving an existence and uniqueness result. The main condition required is that the volatility decays at least linearly close to the boundary, with the remaining conditions on the coefficients being mild. We also characterise the blow-up time for such equations, and demonstrate that it coincides with blow-up of one of the derivatives of the profiles at the shared boundary. The results here extend the work in [10] by incorporating reflection and allowing for drift and volatility coefficients which depend on the spatial variable as well as the solution itself, and are Lipschitz in the solution. The condition on the decay of the volatility at the boundary is also relaxed, and required to be linear only, and we are able to choose a general Lipschitz function h to determine the boundary evolution. The presence of the reflection measure requires us to consider the parabolic obstacle problem, which we prove is a contraction in the space C([0, T ]; H ), where H is the space of continuous functions on (0, 1) which vanish at the endpoints and have derivatives at 0. It also requires us to produce stronger estimates on the heat kernel so that we can bound the L p (Ω ; C([0, T ]; H ))-norms of our processes. We are able to relax the condition on the volatility by formulating the equation in the frame relative to the shared boundary point p(t). Doing so removes the need to consider effects from the boundary shift in our Picard argument, which appear in [10], enabling us to obtain the stronger result. This paper can also be contrasted with [4]. In [4], the volatility coefficients are not required to decay at the shared interface, at the cost of having the boundary derivative given by a function of the profiles in the space of continuous paths. In particular, the classic Stefan boundary condition does not fall within the framework of [4]. The outline for this paper is as follows. We begin in Section 2 by defining our notion of solutions, motivating this by some simple calculations. Section 2 is then concluded by a statement of our main existence and uniqueness theorem. In Section 3 we turn our attention to the deterministic obstacle problem which corresponds to our equations here, and prove a key result which will allow us to ensure that the reflection component will not prevent our SPDE solutions from having derivatives at the boundary. Section 4 is dedicated to establishing some key estimates for the main proof, with some of the technical details deferred to the Appendix. In Section 5 we prove our main result, presenting the arguments for existence and uniqueness for our problem. This is done by first truncating the problem suitably and then performing a Picard iteration, making use of the estimates from Sections 3 and 4. We conclude in Section 6 with simulations of the equations, illustrating the appearance of derivatives when the coefficients fall within our framework and briefly exploring scenarios where this is not the case. 2. Notion of solution and statement of main theorem We begin this section by defining the space H . The solutions to our Stefan problem will be H -valued processes. Definition 2.1. { } H := f ∈ C([0, 1]) | f (0) = f (1) = 0, f ′ (0) exists . Please cite this article as: B. Hambly and J. Kalsi, Stefan problems for reflected SPDEs driven by space–time white noise, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.04.003.

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We equip H with the norm ⏐ ⏐ ⏐ f (x) ⏐ ⏐. ∥ f ∥H := sup ⏐⏐ x ⏐ x∈(0,1] Remark 2.2. (H , ∥ · ∥H ) is a Banach space. In addition, we can characterise H as follows H = { f ∈ C([0, 1]) | ∃g ∈ C([0, 1]) s.t. f (x) = xg(x), g(1) = 0}

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Clearly, for any f ∈ H , the function g such that f (x) = xg(x) is unique, and we have that ∥ f ∥H = ∥g∥∞ .

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Definition 2.3. Let (Ω , F , Ft , P) be a complete filtered probability space. Suppose that W˙ is a space–time white noise defined on this space. Define for A ∈ B(R), Wt (A) := W˙ ([0, t] × A).

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We say that W˙ respects the filtration Ft if (Wt (A))t≥0,A∈B(R) is an Ft - martingale measure i.e. if for every A ∈ B(R), (Wt (A))t≥0 is an Ft -martingale. Let (Ω , F , P) be a complete filtered probability space, and W˙ space–time white noise on R × R. Let Ft be the filtration generated by the white noise, so that Ft = σ ({W (A) | A ∈ B([0, t] × R)}). Suppose that (u 1 , η1 , u 2 , η2 , p) is an Ft -adapted process solving (1.1). Then p : R+ ×Ω ↦→ R is a Ft -adapted process such that the paths of p(t) are C 1 almost surely (note that, in particular, p is Ft -predictable). Let ϕ ∈ Cc∞ ([0, ∞) × (0, 1)), and define the function φ by setting φ(t, x) = ϕ(t, p(t) − x). By multiplying the equation for u 1 in (1.1) by such a φ and integrating over space and time, interpreting the derivatives in the usual weak sense, we obtain that for t ≥ 0, ∫ ∫ ∫ t∫ ∂φ u 1 (t, x)φ(t, x)dx = u 1 (0, x)φ(0, x)dx + u 1 (s, x) (s, x)dxds ∂t R R 0 R ∫ t∫ 2 ∂ φ + u 1 (s, x) 2 (s, x)dxds ∂x 0 R ∫ t∫ + f 1 ( p(s) − x, s, u 1 (s, p(s) − ·))φ(s, x)dxds 0 R ∫ t∫ + σ1 ( p(s) − x, s, u 1 (s, p(s) − ·))φ(s, x)W (dx, ds) 0 R ∫ t∫ + φ(s, x) η1 (ds, d x). +

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We now introduce a change in the spatial variable in order to associate our problem with a fixed boundary problem. Setting v 1 (t, x) = u 1 (t, p(t) − x), the above equation becomes ∫ 1 ∫ 1 v 1 (t, x)φ(t, p(t) − x)dx = v 1 (0, x)φ(0, p(0) − x)dx 0 0 ∫ t∫ 1 ∂φ + v 1 (s, x) (s, p(s) − x)dxds ∂t 0 0 ∫ t∫ 1 ∂ 2φ v 1 (s, x) 2 (s, p(s) − x)dxds + ∂x 0 0 Please cite this article as: B. Hambly and J. Kalsi, Stefan problems for reflected SPDEs driven by space–time white noise, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.04.003.

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∫ t∫

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σ1 (x, s, v 1 (s, ·))φ(s, p(s) − x)W p (dx, ds)

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φ(s, p(s) − x) η1p (ds, d x).

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Here, W˙ p and η1p are obtained from W and η by defining, for t ∈ R+ and A ∈ B(R), ∫ t∫ ∫ t∫ W˙ p ([0, t] × A) = W (ds, dy), η1p ([0, t] × A) = η(dy, ds). 0

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Note that, since the process p(t) is Ft -predictable, W˙ p is then also a space–time white noise which respects the filtration Ft . Also, η1p is a reflection measure for v, so that ∫ T∫ 1 v 1 (t, x) η1p (dt, dx) = 0. 0

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Differentiating φ in time gives

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∂φ ∂ϕ ∂ϕ (t, x) = (t, p(t) − x) + p ′ (t) (t, p(t) − x). ∂t ∂t ∂x It therefore follows that ∫ t∫ 1 ∫ 1 ∫ 1 ∂ϕ v 1 (s, x) (s, x)dxds v 1 (0, x)ϕ(0, x)dx + v 1 (t, x)ϕ(t, x)dx = ∂t 0 0 0 0 ∫ t∫ 1 ∂ϕ v 1 (s, x) p ′ (s) (s, x)dxds + ∂x 0 0 ∫ t∫ 1 2 ∂ ϕ v 1 (s, x) 2 (s, x)dxds + ∂x 0 0 ∫ t∫ 1 f 1 (x, s, v 1 (s, ·))ϕ(s, x)dxds + 0 0 ∫ t∫ 1 + σ1 (x, s, v 1 (s, ·))ϕ(s, x)W p (dx, ds) 0 0 ∫ t∫ 1 + ϕ(s, x) η1p (ds, d x). 0

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W˙ p− ([0, t] × A) = W˙ p ([0, t] × (−A)). Note also that, since (u 1 , η1 , u 2 , η2 , p) is Ft -adapted, we know that (v 1 , η1p , v 2 , η1p , p) is also Ft -adapted. We now define our notion of solution for a class of reflected SPDEs. This will prove useful when defining solutions to our moving boundary problems, and when proving existence for these via a sequence of solutions to truncated versions of the equations. Definition 2.4. Let (Ω , F , Ft , P) be a complete filtered probability space. Let W˙ be a space– time white noise on this space which respects the filtration Ft . Suppose that v˜ is a continuous Ft -adapted process taking values in H . Let F : H → H and h : H × H → H . For the Ft -stopping time τ , we say that the pair (v, η) is a local solution to the reflected SPDE ∂ F(v) ∂v = ∆v + h(v(t, ·), v(t, ˜ ·)) + f (x, t, v(t, ·)) + σ (x, t, v(t, ·))W˙ + η ∂t ∂x on [0, ∞) × [0, 1] with Dirichlet boundary conditions v(t, 0) = v(t, 1) = 0 and initial data v0 ∈ H , with v0 ≥ 0, until time τ , if (i) (ii) (iii) (iv) (v)

For every x ∈ [0, 1] and every t ≥ 0, v(t, x) is an Ft -measurable random variable. v ⏐≥ 0 almost surely. v ⏐[0,τ )×[0,1] ∈ C([0, τ ); H ) almost surely. v(t, x) = ∞ for every t ≥ τ and x ∈ [0, 1] almost surely. η is a measure on (0, 1) × [0, ∞) such that (a) For every measurable map ψ : [0, 1] × [0, ∞) → R, ∫ t∫ 1 ψ(x, s) η(dx, ds) 0

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∫is∞F∫t -measurable. 1 (b) 0 0 v(t, x) η(dx,dt) = 0. (vi) There exists a localising sequence of stopping times τn ↑ τ almost surely, such that for every ϕ ∈ Cc1,2 ([0, ∞) × [0, 1]) such that ϕ(s, 0) = ϕ(s, 1) = 0 for every s ∈ [0, ∞), ∫ 1 ∫ 1 ∫ t∧τn ∫ 1 ∂ϕ v(t ∧ τn , x)ϕ(t, x)dx = v(0, x)ϕ(0, x)dx + v(s, x) (s, x)dxds ∂t 0 0 0 0 ∫ t∧τn ∫ 1 ∂ 2ϕ + v(s, x) 2 (s, x)dxds ∂x 0 0 ∫ t∧τn ∫ 1 ∂ϕ − h(v(s, ·), v(s, ˜ ·))F(v(s, x)) (s, x)dxds ∂x 0 0 ∫ t∧τn ∫ 1 f (x, s, v(s, ·))ϕ(s, x)dxds + 0

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Please cite this article as: B. Hambly and J. Kalsi, Stefan problems for reflected SPDEs driven by space–time white noise, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.04.003.

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∫

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ϕ(s, x) η(ds, d x).

(2.1)

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for every t ≥ 0 almost surely. We say that a local solution is maximal if there does not exist a solution to the equation on a larger stochastic interval, and we say that a local solution is global if we can take τn = ∞ in (2.1). The following definitions provide us with notation which will allow us to easily move between the relative frame (measured with respect to the current position of the boundary) and the fixed frame when discussing solutions to our equations. Definition 2.5. For a function p : [0, T ] → R we define θ p1 : [0, T ] × R → [0, T ] × R such that θ p1 (t, x) := (t, p(t) − x). We similarly define θ p2 : [0, T ] × R → [0, T ] × R such that θ p2 (t, x) := (t, x − p(t)). Definition 2.6. For a space–time white noise W˙ , we denote by W˙ − the space–time white noise such that W˙ − ([0, t] × A) = W˙ ([0, t] × (−A)). Before finally stating our definition for solutions to (1.1), we first outline the conditions on our coefficients f i , σi and h. (i) For i = 1, 2, f i , σi : [0, 1] × [0, ∞) × H → R are measurable mappings. (ii) h : H × H → R is a measurable map. (iii) For i = 1, 2, f i is such that for every T > 0 there exists C T such that for every t ∈ [0, T ] and every x ∈ [0, 1], | f i (x, t, 0)| ≤ C T . (iv) For i = 1, 2, f i satisfies the local Lipschitz condition that for every M, T > 0, there exists C T,M such that for every t ∈ [0, T ], every x ∈ [0, 1] and every u, v ∈ H with ∥u∥H , ∥v∥H ≤ M, | f i (x, t, u) − f i (x, t, v)| ≤ C T,M ∥u − v∥H . (v) For i = 1, 2, σi is such that for every T > 0 there exists C T such that for every t ∈ [0, T ] and every x ∈ [0, 1] |σi (x, t, 0)| ≤ C T x (vi) For i = 1, 2, σi satisfies the local Lipschitz condition that for every M, T > 0, there exists C T,M such that for every t ∈ [0, T ], every x ∈ [0, 1] and every u, v ∈ H with ∥u∥H , ∥v∥H ≤ M, |σi (x, t, u) − σi (x, t, v)| ≤ C T,M x∥u − v∥H . Please cite this article as: B. Hambly and J. Kalsi, Stefan problems for reflected SPDEs driven by space–time white noise, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.04.003.

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(vii) h is bounded on bounded sets. (viii) h satisfies the local Lipschitz condition that for every M > 0, there exists C M such that for every u, v ∈ H with ∥u∥H , ∥v∥H ≤ M, |h(u 1 , v1 ) − h(u 2 , v2 )| ≤ C M (∥u 1 − u 2 ∥H + ∥v1 − v2 ∥H ).

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Remark 2.7. An immediate consequence of the conditions on f i and σi is that they satisfy local linear growth conditions. That is, for every T, M > 0 there exists a constant C T,M such that for every t ∈ [0, T ], every x ∈ [0, 1] and every u ∈ H with ∥u∥H ≤ M,

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Remark 2.8. The class of permitted functions for h includes as a subclass functions of the form ) ( ∂v ∂u (0), (0) , h(u, v) = h˜ ∂x ∂x where h˜ is a Lipschitz function from R × R → R. Remark 2.9. The x term in the linear growth and Lipschitz conditions for σ ensure that the volatility will decay at least linearly as we approach the boundary i.e. as x ↓ 0. We note that, in the case when σ (x, t, u) = σ˜ (x, t, u(x)), the typical Lipschitz condition |σ˜ (x, t, u) − σ˜ (x, t, v)| ≤ C|u − v| implies the Lipschitz condition (vi). In particular, volatility functions of the form σ (x, t, u) = σ1 (x) + σ2 (x)u(x), where

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We are now in position to state our definition for a solution to problem (1.1). Following this, we state the main result, namely existence and uniqueness for maximal solutions to these equations. Definition 2.10. Let (Ω , F , Ft , P) be a complete filtered probability space. Let W˙ be a space– time white noise on this space which respects the filtration Ft . We say that the quintuple (u 1 , η1 , u 2 , η2 , p) satisfies the reflected stochastic Stefan problem ∂u 1 = ∆u 1 + f 1 ( p(t) − x, t, u 1 (t, p(t) − ·)) + σ1 ( p(t) − x, t, u 1 (t, p(t) − ·))W˙ + η1 ∂t ∂u 2 = ∆u 2 + f 2 (x − p(t), t, u 2 (t, p(t) + ·)) + σ2 (x − p(t), t, u 2 (t, p(t) + ·))W˙ + η2 , ∂t p ′ (t) = h(u 1 (t, p(t) − ·), u 2 (t, p(t) + ·)), with initial data (u 10 , u 20 ) ∈ H 2 , where u 10 , u 20 ≥ 0, up to the Ft -stopping time τ if Please cite this article as: B. Hambly and J. Kalsi, Stefan problems for reflected SPDEs driven by space–time white noise, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.04.003.

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(i) (v 1 , η˜ 1 ) := (u 1 ◦ (θ p1 )−1 , η1 ◦ (θ p1 )−1 ) solves, in the sense of Definition 2.4, the reflected SPDE ∂v ∂v = ∆v 1 − h(v 1 (t, ·), v 2 (t, ·)) + f 1 (x, t, v 1 (t, ·)) + σ1 (x, t, v 1 (t, ·))W˙ + η˜ 1 ∂t ∂x

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with Dirichlet boundary conditions v 1 (0) = v 1 (1) = 0 and initial data v 1 (0, x) = u 10 (x) until time τ . (ii) (v 2 , η˜ 2 ) := (u 2 ◦ (θ p2 )−1 , η2 ◦ (θ p2 )−1 ) solves, in the sense of Definition 2.4, the reflected SPDE

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with Dirichlet boundary conditions v 2 (0) = v 2 (1) = 0 and initial data v 2 (0, x) = u 20 (x) until time τ . (iii) p ′ (t) = h(v 1 (t, ·), v 2 (t, ·)).

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Theorem 2.11. problem.

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There exists a unique maximal solution to the reflected stochastic Stefan

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Remark 2.12. It is immediate from the definition that existence of a unique maximal solution to the reflected stochastic Stefan problem is equivalent to the existence of a unique maximal solution to the system of coupled reflected SPDEs

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∂v 1 ∂v 1 = ∆v 1 − h(v 1 (t, ·), v 2 (t, ·)) + f 1 (x, t, v 1 (t, ·)) + σ1 (x, t, v 1 (t, ·))W˙ + η1 , ∂t ∂x ∂v 2 ∂v 2 = ∆v 2 + h(v 1 (t, ·), v 2 (t, ·)) + f 2 (x, t, v 2 (t, ·)) + σ2 (x, t, v 2 (t, ·))W˙ − + η2 . ∂t ∂x (2.2) 3. The deterministic obstacle problem and the corresponding bounds in H We begin this section by discussing some relevant work on a deterministic obstacle problem. The obstacle problem in the form given here was originally discussed in [8]. The space C0 ((0, 1)) here denotes the space of continuous functions on (0, 1) which vanish at the endpoints. Definition 3.1. Let z ∈ C([0, T ] × [0, 1]) with z(t, ·) ∈ C0 ((0, 1)) for every t ∈ [0, T ] and z(0, ·) ≤ 0. We say that the pair (w, η) satisfies the heat equation with obstacle z if: (i) w ∈ C([0, T ] × [0, 1]), w(t, 0) = w(t, 1) = 0, w(0, x) ≡ 0 and w ≥ z. (ii) η is a measure on (0, 1) × [0, T ]. (iii) w weakly solves the PDE ∂w ∂ 2w = +η ∂t ∂x2 Please cite this article as: B. Hambly and J. Kalsi, Stefan problems for reflected SPDEs driven by space–time white noise, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.04.003.

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That is, for every t ∈ [0, T ] and every φ ∈ C 2 ((0, 1)) ∩ C0 ((0, 1)), ∫ 1 ∫ t∫ 1 ∫ t∫ 1 w(t, x)φ(x)dx = w(s, x)φ ′′ (x)dxds + φ(x) η(dx, ds).

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(iv)

∫t ∫1 0

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(w(s, x) − z(s, x)) η(dx, ds) = 0.

Existence and uniqueness for solutions to this problem was proved in [8]. It was also shown that the difference between two solutions can be bounded in the L ∞ -norm by the difference in the L ∞ -norm of the obstacles. We now adapt this result, and show that solutions to the obstacle problem in the case when the obstacle lies in C([0, T ]; H ) are themselves in the space C([0, T ]; H ). We also prove the corresponding estimate, that we can control the H norm of the difference in the solutions by the H -norm of the difference of the obstacle. This will be required to prove existence for our reflected SPDEs via a Picard argument. Theorem 3.2. Suppose that z 1 , z 2 ∈ C([0, T ]; H ). Let w1 , w2 solve the corresponding parabolic obstacle problems with obstacles z 1 and z 2 . Then we have that w1 , w2 ∈ C([0, T ]; H ) and ∥w1 − w2 ∥H ,T ≤ ∥z 1 − z 2 ∥H ,T . Proof. Define φ(t) := ∥z 1 − z 2 ∥H ,t . For i = 1, 2 and ϵ > 0, let wiϵ solve the following penalised equations ∂wiϵ = ∆wiϵ + gϵ (wiϵ − z), ∂t with Dirichlet boundary conditions wiϵ (t, 0) = wiϵ (t, 1) = 0 and zero initial data, where we define gϵ (x) :=

1 arctan((x ∧ 0)2 ). ϵ

Then we know that wiϵ ↑ wi for i = 1, 2 (see, for example, the proof of Theorem 1.4 in [8]). We define k ϵ (t, x) := w1ϵ − w2ϵ − xφ(t). Then k ϵ ∈ C([0, T ]; H 1 (0, 1)) is a weak solution to the equation ∂k ϵ dφ = ∆k ϵ + gϵ (w1ϵ − z 1 ) − gϵ (w2ϵ − z 2 ) − x , ∂t dt with boundary conditions k(t, 0) = 0, k(t, 1) = −φ(t) and negative initial data k(0, x) = −x∥z 1 (0, ·) − z 2 (0, ·)∥H . Note that, since φ is increasing in t, the weak derivative of φ is positive. Testing this equation with (k ϵ )+ and integrating over space and time, we obtain that ∫ T∫ 1 ∫ T ∫ 1 ⏐ ϵ + ⏐2 ⏐ ∂(kt ) ⏐ 1 ϵ + 2 1 ⏐ ⏐ dxdt + ∥(k T ) ∥ L 2 = ∥(k0ϵ )+ ∥2L 2 − gϵ (w1ϵ (t, x) ⏐ ∂x ⏐ 2 2 0 0 0 0 − z 1 (t, x))(k ϵ (t, x))+ dxdt ∫ T∫ 1 (3.1) − gϵ (w2ϵ (t, x) − z 2 (t, x))(k ϵ (t, x))+ dxdt 0 0 ∫ T∫ 1 − x(k ϵ (t, x))+ dx dφt . 0

0

Please cite this article as: B. Hambly and J. Kalsi, Stefan problems for reflected SPDEs driven by space–time white noise, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.04.003.

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Notice that gϵ is a decreasing function. Also, on the set k ϵ ≥ 0, w1ϵ + z 1 ≥ w2ϵ + z 2 . It then follows that the right hand side of (3.1) is negative. Therefore, (k ϵ )+ = 0, and so w1ϵ

−

w2ϵ

≤ xφ(t). w1ϵ

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3

w2ϵ ,

and we obtain that, for every ϵ > 0, ⏐ ϵ ⏐ ⏐ w1 (t, x) w2ϵ (t, x) ⏐ ⏐ ≤ φ(T ). ⏐ − sup sup ⏐ ⏐ x x t∈[0,T ] x∈(0,1]

Interchanging

1

Letting ϵ ↓ 0, we deduce that ⏐ ⏐ ⏐ w1 (t, x) w2 (t, x) ⏐ ⏐ ⏐ ≤ φ(T ). sup sup ⏐ − ⏐ x x t∈[0,T ] x∈(0,1]

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(3.2)

We now argue that the functions wi ∈ C([0, T ]; H ). Note that we have C([0, T ]; H ) regularity for solutions to the obstacle problem provided that the obstacle is smooth (see, for example, Theorem 4.1 in [1]). Let z n be a sequence of smooth obstacles such that z n → z in C([0, T ]; H ), and denote by wn the solution to the obstacle problem with obstacle z n . Then, by (3.2), the sequence wn is Cauchy in C([0, T ]; H ), and so converges in C([0, T ]; H ) to some w. ˜ We also have that ⏐ ⏐ ⏐ wn (t, x) w(t, x) ⏐ ⏐ ⏐ → 0. sup sup ⏐ − x x ⏐

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t∈[0,T ] x∈(0,1]

Therefore, we have that w˜ = w, and so w ∈ C([0, T ]; H ). This concludes the proof.

□

4. Key estimates

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In order to prove existence for our reflected SPDE, we will perform a Picard iteration in the space L p (Ω ; C([0, T ]; H )) to show that solutions of (2.2) exist. Having defined (v 1,n , η1,n , v 2,n , η2,n ), we will define our (n + 1)th approximation of a solution to be given by the solutions of ∂v

∂v p ′ (t) + σ (x, t, v 1,n (t, ·))W˙ + η1,n+1 , ∂t ∂ x n+1 ∂v 2,n+1 ∂v 2,n ′ =∆v 2,n+1 + f (x, t, v 2,n (t, ·)) + p (t) ∂t ∂ x n+1 2,n − 2,n+1 + σ (x, t, v (t, ·))W˙ + η . ∫ t ( 1,n ) pn+1 (t) = p(0) + h v (s, ·), v 2,n (s, ·) ds. 0 [ ] p In order to control the difference E ∥v 1,n+1 − v 1,n ∥H ,T , we will use the result from Section 3. This gives [ ] [ ] p p E ∥v 1,n+1 − v 1,n ∥H ,T ≤ C p E ∥z 1,n+1 − z 1,n ∥H ,T , 1,n+1

=∆v 1,n+1 + f (x, t, v 1,n (t, ·)) −

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1,n

1,n+1

where z solves 1,n+1 ∂z ∂v 1,n ′ ∂2W = ∆z 1,n+1 − pn+1 (t) + f (x, t, v 1,n (t, ·)) + σ (x, t, v 1,n (t, ·)) . ∂t ∂x ∂ x∂t [ 2,n+1 ] [ ] p p We deduce in the same way that E ∥v − v 2,n ∥H ,T ≤ C p E ∥z 2,n+1 − z 2,n ∥H ,T , where z 2,n+1 solves ∂z 2,n+1 ∂v 2,n ′ ∂2W = ∆z 2,n+1 + pn+1 (t) + f (x, t, v 2,n (t, ·)) + σ (x, t, v 2,n (t, ·)) . ∂t ∂x ∂ x∂t Please cite this article as: B. Hambly and J. Kalsi, Stefan problems for reflected SPDEs driven by space–time white noise, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.04.003.

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By writing this in mild form, noting that the mild and weak forms are equivalent for such equations, we see that we will require estimates on the terms (i) 1

∫ t∫ 3

0

0

[ ] 1 G(t − s, x, y) f (x, s, v i,n (s, ·)) − f (x, s, v i−1,n (s, ·)) dyds. x

(4.1)

(ii) 1

∫ t∫ 4

0

0

[ ] y σ (x, s, v i,n (s, ·)) − σ (x, s, v i−1,n (s, ·)) G(t − s, x, y) W (dy, ds). (4.2) x y

(iii)

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[ i,n ] v (s, y) n ′ v i,n−1 (s, y) n−1 ′ y ∂G (t − s, x, y) ( p ) (s) − ( p ) (s) dyds. (4.3) y y 0 0 x ∂y Obtaining such estimates will be the focus in this section.

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∫ t∫

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In this section we will state and prove the heat kernel estimates which will be crucial in proving existence and uniqueness for our equations. In particular, they allow us to show that solutions to certain SPDEs lie in the space C([0, T ]; H ), and also enable us to obtain estimates in L p (Ω ; C([0, T ]; H )) for these solutions. We define G to be the Dirichlet heat kernel on [0, 1], so ( ) ( )] ∞ [ ∑ 1 (x − y + 2n)2 (x + y + 2n)2 G(t, x, y) := √ exp − − exp − . 4t 4t 4πt n=−∞ We then define for t > 0, y ∈ [0, 1] and x ∈ (0, 1] ˜ x, y) := y G(t, x, y). G(t, x For t > 0, y ∈ [0, 1] and x = 0, we set ˜ x, y) := y ∂G (t, 0, y). G(t, ∂x We also define y ∂G H˜ (t, x, y) := (t, x, y) x ∂y for t ∈ [0, T ], x ∈ (0, 1] and y ∈ [0, 1], and set

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∂2G (t, 0, y) H˜ (t, 0, y) := y ∂ y∂ x for t ∈ [0, T ] and y ∈ [0, 1].

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Remark 4.1.

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Note that [ ( ) ( )] 1 (x − y)2 (x + y)2 G(t, x, y) = √ exp − − exp − 4t 4t 4π t [ ( ) ( )] 1 (x − y − 2)2 (x + y − 2)2 + √ exp − − exp − + L(t, x, y), 4t 4t 4π t

Please cite this article as: B. Hambly and J. Kalsi, Stefan problems for reflected SPDEs driven by space–time white noise, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.04.003.

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where L is a smooth function on [0, T ] × [0, 1] × [0, 1], vanishing on t = 0 and x = 0, 1. Since the arguments for bounding the second term on the right hand side are similar to those for bounding the first, and estimates for L are simple due to L being smooth, it is sufficient to prove the estimates for the first term, ( ) )] [ ( (x + y)2 (x − y)2 1 − exp − . exp − G 1 (t, x, y) := √ 4t 4t 4π t Proposition 4.2.

The following estimates hold for G, G˜ and H˜ .

1. For T > 0, ∃ C T > 0 such that for every t ∈ [0, T ] ⏐ ∫ 1⏐ ⏐1 ⏐ CT ⏐ ⏐ sup ⏐ x G(t, x, y)⏐ dy ≤ √t . x∈(0,1] 0 2. For T > 0, ∃ C T > 0 such that for every t ∈ [0, T ], ∫ 1 T ˜ x, y)2 dy ≤ C sup G(s, √ . t x∈[0,1] 0 3. For T > 0 and q ∈ (1, 2), ∃ C T,q > 0 such that ∫ t [∫ 1 ( )2 ]q ˜ ˜ sup G(s, x, z) − G(t, y, z) dy ds ≤ C T,q |x − y|(2−q)/3 . t∈[0,T ] 0

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4. For T > 0 and q ∈ (1, 2), ∃ C T,q > 0 such that for 0 ≤ s ≤ t ≤ T ]q ∫ s [∫ 1 ˜ − r, x, y) − G(s ˜ − r, x, y))2 dy dr ≤ C T,q |t − s|(2−q)/2 . sup (G(t x∈[0,1] 0

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Proof. The proof of inequality (2) can be found in the proof of Lemma 3.2.1 in [10]. The other inequalities can be shown by adapting some of the arguments from [10], and their proofs are deferred to the Appendix. □

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We would now like to translate the estimates on the heat kernel into estimates on the terms (4.1), (4.2) and (4.3) appearing in the mild form for the problem. We would also like to ensure continuity of these terms. In the case of the drift and moving boundary terms in the mild form, this is essentially an immediate consequence of the heat kernel estimates. For the stochastic term, more work is required, as we need to control the L p -norm of the supremum of the term over the space–time interval [0, T ] × [0, 1]. We can achieve this by a standard application of the Garsia–Rodemich–Rumsey Lemma. Please cite this article as: B. Hambly and J. Kalsi, Stefan problems for reflected SPDEs driven by space–time white noise, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.04.003.

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Proposition 4.3. Let f ∈ L p (Ω ; L ∞ ([0, T ] × [0, 1])). Define F(t, x) such that, for t ∈ [0, T ] and x ∈ (0, 1], ∫ t∫ 1 G(t − s, x, y) f (s, y)dydy. F(t, x) :=

4

Then we have that, for p > 2,

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1. F ∈ C([0, T ]; H ) almost surely. 2. ∫ t [ ] [ ] p p E ∥F∥H ,T ≤ C p,T E ∥ f ∥∞,s ds. 0

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Proof. F ∈ C([0, T ]; H ) is clear, since F is simply the solution to the heat equation with Dirichlet boundary conditions at x = 0, 1, initial data 0 and source term f . For the second part, we note that ⏐ p ⏐∫ t (∫ 1 ⏐p ⏐∫ t ∫ 1 ) ⏐ ⏐ ⏐ ⏐ 1 1 ⏐ ⏐ ⏐ G(t − s, x, y) f (s, y)dyds ⏐ ≤ ⏐ G(t − s, x, y)dy ∥ f ∥∞,s ds ⏐⏐ ⏐ 0 0 x 0 0 x [∫ (∫ )q ] p/q t 1 1 ≤ G(t − s, x, y)dy ds 0 0 x ∫ t p ∥ f ∥∞,s ds, × 0

(4.4)

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where q = p/( p − 1). By estimate (1) in Proposition 4.2, we have that [∫ (∫ )q ] p/q )q ] p/q [∫ t ( t 1 1 1 ds = C T, p . G(t − s, x, y)dy ds ≤ C T, p √ t −s 0 0 x 0 Therefore, taking the supremum over x ∈ (0, 1] on the left hand side of (4.4) and then taking expectations of both sides gives the result. □ Proposition 4.4. Let f ∈ L p (Ω ; L ∞ ([0, T ]; H )). Define J (t, x) so that ∫ t∫ 1 ∂G (t − s, x, y) f (s, y)dyds. J (t, x) := 0 0 ∂y Then we have that 1. J ∈ C([0, T ]; H ) almost surely. 2. ∫ t [ ] [ ] p p E ∥J ∥H ,T ≤ C T, p E ∥ f ∥H ,s ds. 0

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Proof. We have that, for x ∈ (0, 1], ∫ t∫ 1 J (t, x) y ∂G f (s, y) := (t − s, x, y) dyds x y 0 0 x ∂y ∫ t∫ 1 y ∂G f (t − s, y) = (s, x, y) dyds. y 0 0 x ∂y Please cite this article as: B. Hambly and J. Kalsi, Stefan problems for reflected SPDEs driven by space–time white noise, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.04.003.

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is well defined by Note that, by the estimates in Proposition A.6 in [4], we know that J (t,x) x our integral expression and continuous on [0, T ] × (0, 1]. So it is left to show that J (t,x) can be x extended to a continuous function on [0, T ] × [0, 1]. In the following arguments, we focus on the G 1 component of G (see Remark 4.1). Suppose that (tn , xn ) is a sequence in [0, T ] × [0, 1] converging to (t, 0). Note that ⏐ ⏐ ∫ tn ∫ 1 f (tn − s, y) ⏐⏐ y ⏐⏐ ∂G 1 (s, xn , y) 1{y<2xn } ⏐ ⏐ dyds xn ∂ y y 0 0 ⏐ ⏐ ∫ tn ∫ 1 ⏐ ∂G 1 ⏐ ⏐ ≤2∥ f ∥H ,T 1{y<2xn } ⏐ (s, xn , y)⏐⏐ dyds. ∂y 0 0 For p ∈ (1, 2), we have that this is at most ⏐p )1/ p (∫ tn ∫ 1 )1/q (∫ tn ∫ 1 ⏐ ⏐ ⏐ ∂G 1 ⏐ ⏐ (s, x , y) dyds . (4.5) 2∥ f ∥H ,T 1{y<2xn } dyds × n ⏐ ⏐ ∂y 0 0 0 0 Since ⏐p ∫ t∫ 1⏐ ⏐ ∂G 1 ⏐ ⏐ ⏐ dyds < ∞, sup sup (s, x, y) ⏐ ∂y ⏐ t∈[0,T ] x∈[0,1] 0

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we have that (4.5) converges to zero as n → ∞. Now note that

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y ∂G 1 f (tn − s, y) ∂ G f (t − s, y) (s, xn , y) → 1{s∈[0,t]} y (s, 0, y) xn ∂ y y ∂ y∂ x y almost everywhere in [0, T ] × [0, 1]. In addition, we have that for y ≥ 2xn , ⏐ ⏐ ⏐ )⏐⏐ √ ⏐ y ∂G 1 ⏐ ⏐ y ( −(xn −y)2 /4s −(xn +y)2 /4s ⏐ ⏐ ⏐ ⏐ 4π ⏐ (s, xn , y)⏐ = ⏐ + (xn + y)e √ (xn − y)e ⏐ xn ∂ y x s s ⏐ n ( ⏐ ) ⏐ y ⏐ 2 2 ≤ ⏐⏐ √ e−(xn −y) /4s + e−(xn +y) /4s ⏐⏐ s s ⏐ )⏐⏐ ⏐ y 2 ( −(x −y)2 /4s −(xn +y)/4s ⏐ n ⏐ +⏐ −e (4.6) √ e ⏐ xn s s ⏐ ⏐ )⏐ ⏐ y ( 2 2 ≤ ⏐⏐ √ e−(xn −y) /4s + e−(xn +y) /4s ⏐⏐ s s ⏐ ⏐ ⏐ y 2 −(x −y)2 /4s ( )⏐ 1 − e−xn y/s ⏐⏐ . + ⏐⏐ √ e n xn s s Since y ≥ 2xn ≥ 0, we have that 2

1{y≥2xn ,s∈[0,tn ]}

e

−(xn −y)2 /4s

≤e

−y 2 /16s

.

Note also that, for x ≥ 0, 1 − e−x ≤ x. Therefore, we can bound (4.6) by ⏐ ⏐ )⏐⏐ ⏐⏐ y 2 ⏐ y ( −y 2 /16s xn y ⏐⏐ −y 2 /4s ⏐ −y 2 /16s ⏐ ⏐ √ e +e × ⏐s s ⏐ + ⏐ x s √s e s ⏐ n ( ) y y3 2 2 ≤ C √ e−y /16s + 2 √ e−y /16s . s s s s We can similarly bound the other components of G, to obtain that, for every n, ⏐ ⏐ ( ) √ ⏐ y ∂G ⏐ y −y 2 /16s y 3 −y 2 /16s ⏐ ⏐ 4π ⏐ (s, xn , y)⏐ ≤ C √ e + 2√ e . x ∂y s s s s

12

13

14

15

16

17

(4.7)

n

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It follows that, for every n, ⏐ ⏐ ⏐ ⏐ ⏐1{y≥2x ,s∈[0,t ]} y ∂G (s, xn , y) f (tn − s, y) ⏐ n n ⏐ ⏐ xn ∂ y y ( ) y y3 2 2 ≤ C √ e−y /16s + 2 √ e−y /16s ∥ f ∥H ,T s s s s 1

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This function is integrable, and so we can apply the DCT to obtain that ∫ tn ∫ 1 y ∂G f (s, y) 1{y≥2xn } (tn − s, xn , y) dyds xn ∂ y y 0 0 ∫ tn ∫ 1 y ∂G f (tn − s, y) 1{y≥2xn } = (s, xn , y) dyds xn ∂ y y 0 0 ∫ t∫ 1 2 ∂ G → (s, 0, y) f (t − s, y)dyds 0 0 ∂ y∂ x ∫ t∫ 1 2 ∂ G = (t − s, 0, y) f (s, y)dyds. 0 0 ∂ y∂ x Another application of the DCT (note that we can use the same dominating function as in (4.7)) gives that, if tn is a sequence in [0, T ] and tn → t, then ∫ tn ∫ 1 2 ∫ t∫ 1 2 ∂ G ∂ G (tn − s, 0, y) f (s, y)dyds → (t − s, 0, y) f (s, y)dyds. ∂ y∂ x ∂ y∂ x 0 0 0 0 So we have shown that J ∈ C([0, T ]; H ) almost surely. We now prove the bound (2). For p > 2, we have that ⏐ ⏐ (∫ t [∫ 1 ⏐ ⏐ ]q )1/q (∫ t )1/ p ⏐ y ∂G ⏐ J (t, x) ⏐ ⏐ p ⏐ ⏐ ⏐≤ ⏐ dy ds × ∥ f ∥ ds . (s, x, y) H ,s ⏐ x ∂y ⏐ x ⏐ ⏐ 0 0 0 Arguing as before, considering the cases y < 2x and y ≥ 2x separately, we have that ⏐ ⏐ ⏐ (⏐ ) ⏐ y ∂G ⏐ ⏐ ∂G ⏐ y −y 2 /16s y 3 −y 2 /16s ⏐ ⏐ ⏐ ⏐ + 2√ e . ⏐ x ∂ y (s, x, y)⏐ ≤ C ⏐ ∂ y (s, x, y)⏐ + s √s e s s For p > 2 and q = p/( p − 1) ∈ (1, 2), and we have that ⏐ ) ]q ∫ t [∫ 1 (⏐ ⏐ ∂G ⏐ y 3 −y 2 /16s y −y 2 /16s ⏐ ⏐ + 2√ e dy ds ⏐ ∂ y (s, x, y)⏐ + s √s e s s 0 0 )q ∫ t( 1 ≤C ds = C T, p < ∞. √ s 0

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It follows that, for p > 2, ⏐ ⏐ (∫ t )1/ p ⏐ J (t, x) ⏐ p ⏐ ⏐ ≤ C T, p ∥ f ∥H ,s ds . ⏐ x ⏐ 0

13

Therefore,

14

sup

⏐ ⏐ ∫ t ⏐ J (t, x) ⏐ p p ⏐ ≤ C T, p sup ⏐⏐ ∥ f ∥H ,s ds. x ⏐

t∈[0,T ] x∈(0,1] 15

0

The result then follows by taking expectations. □ Please cite this article as: B. Hambly and J. Kalsi, Stefan problems for reflected SPDEs driven by space–time white noise, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.04.003.

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Proposition 4.5. Suppose that p > 8 and let f ∈ L p (Ω ; L ∞ ([0, T ]; H )). Define K (t, x) such that, for t ∈ [0, T ] and x ∈ [0, 1], ∫ t∫ 1 K (t, x) := G(t − s, x, y) f (s, y)W (dy, ds). 0

1 2

3

0

Then we have that

4

1. K ∈ C([0, T ]; H ) almost surely. 2. ∫ t [ ] [ ] p p E ∥K ∥H ,t ≤ C T, p E ∥ f ∥H ,s ds.

5

6

0

Proof. In this case, we will need to apply the Garsia–Rodemich–Rumsey Lemma to obtain continuity of K (t,x) , and to suitably control the supremum of this process. Define for t ∈ [0, T ] x and x ∈ [0, 1] ∫ t∫ 1 ˜ − s, x, y) f (s, y) W (dy, ds). G(t L(t, x) := y 0 0 Note that, for t ∈ [0, T ] and x ∈ (0, 1], we then have that L(t, x) = K (t, x)/x. For x, z ∈ (0, 1] and s ≤ t ∈ [0, T ], we have [ ⏐∫ ∫ ⏐p ] ⏐ ⏐ t 1 [ ] f (r, y) p ˜ − r, x, y) W (dy, dr )⏐⏐ E |L(t, x) − L(s, z)| ≤E ⏐⏐ G(t y s 0 [⏐∫ s ∫ 1 [ ] ⏐ (4.8) ˜ − r, x, y) − G(s ˜ − r, z, y) G(t + E ⏐⏐ 0 0 ⏐p] ⏐ f (r, y) × W (dy, dr )⏐⏐ . y

7 8 9

10

11 12

13

Applying Burkholder’s inequality gives that the first term on the right hand side is at most ⎡⏐ ⏐ p/2 ⎤ )2 ⏐∫ t ∫ 1 ( ⏐ f (r, y) ⏐ ⏐ ˜ − r, x, y) E ⎣⏐ G(t dydr ⏐ ⎦ ⏐ s 0 ⏐ y [⏐∫ (∫ ⏐ p/2 ] ) 1 ⏐ t ⏐ ˜ − r, x, y)2 dy ∥ f ∥2H ,r dr ⏐ ≤ E ⏐⏐ G(t . ⏐ s

0

An application of H¨older’s inequality then bounds this by [∫ (∫ ) p/( p−2) ]( p−2)/2 ∫ t t 1 [ ] p 2 ˜ G(t − r, x, y) dy dr × E ∥ f ∥H ,r dr . s

0

14

15

s

By estimate (2) in Proposition 4.2, this can be bounded by ∫ t [ ] p ( p−4)/4 C|t − s| × E ∥ f ∥H ,r dr .

16

17

s

Arguing similarly for the second term in (4.8), making use of the other estimates (3) and (4) in Proposition 4.2, we see that ∫ t [ ] ( ) p−4 [ ] p E |L(t, x) − L(s, z)| p ≤ C T, p |t − s|1/4 + |x − z|1/6 × E ∥ f ∥H ,r dr. 0

Please cite this article as: B. Hambly and J. Kalsi, Stefan problems for reflected SPDEs driven by space–time white noise, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.04.003.

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1

Let d be the metric on [0, T ] × [0, 1] given by

2

d((t, x), (s, y)) := |t − s|1/4 + |x − y|1/6 .

3 4

5

6

7 8

9

10

11

12

13 14 15 16

17

18

19 20 21

22

23

24 25

26

Applying the Garsia–Rodemich–Rumsey Lemma in the form of Corollary A.3 from [2], with respect to this metric, we see that there exists a random variable X ≥ 0 such that ∫t [ ] p 1. E [X p ] ≤ C T, p 0 E ∥ f ∥H ,r dr . 2. ( )( p−8)/ p |L(t, x) − L(s, y)| ≤ X |t − s|1/4 + |x − y|1/6 In particular, we see that L is continuous on [0, T ]×[0, 1] almost surely, from which it follows that K ∈ C([0, T ]; H ) almost surely. In addition, we have that for t ∈ [0, T ] and x ∈ [0, 1] |L(t, x)| = |L(t, x) − L(0, x)| ≤ C T, p × X. It follows that ∫ [ ] [ ] p p E ∥K ∥H ,T = E ∥L∥∞,T ≤ C T, p

T 0

[ ] p E ∥ f ∥H ,t dt. □

5. Existence and uniqueness We are now in position to prove existence and uniqueness for our Stefan problem. As mentioned in the introduction, we prove existence via a Picard argument for a truncated version of the problem. Existence for the main problem is then deduced by concatenation. Before doing so, the following truncation map is introduced. Definition 5.1. For M > 0, we define the map FM : H → H such that ] ⎧ [ ⎨x u(x) ∧ M x ∈ (0, 1] x FM (u)(x) := ⎩ 0 x =0 Remark 5.2. FM (u) is a well-defined map from H to H i.e. FM (u) ∈ H for u ∈ H . Continuity on (0, 1] is clear, and we have defined FM (u)(0) = 0. We therefore only need to argue the existence of a derivative for FM (u) at x = 0. Note that, for x > 0, [ ] u(x) FM (u)(x) = ∧M . x x This converges as x ↓ 0, since

u(x) x

converges.

Theorem 5.3. Fix some M > 0. Then there exists a unique solution (v 1M , η1M , v 2M , η2M ) to the system of coupled SPDEs ∂v 1M ∂(FM (v 1M )) = ∆v 1M − p ′ (t) + f 1,M (x, t, v 1M (t, ·)) + σ1,M (x, t, v 1M (t, ·))W˙ + η1M , ∂t ∂x ∂v 2M ∂(FM (v 2M )) = ∆v 2M + p ′ (t) + f 2,M (x, t, v 2M (t, ·)) + σ2,M (x, t, v 2M (t, ·))W˙ − + η2M , ∂t ∂x p ′ (t) = h M (v 1M (t, ·), v 2M (t, ·)) Please cite this article as: B. Hambly and J. Kalsi, Stefan problems for reflected SPDEs driven by space–time white noise, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.04.003.

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where W˙ is space–time white noise with respect to the complete probability space (Ω , F , Ft , P), v 1M (t, 0) = v 1M (t, 1) = v 2M (t, 0) = v 2M (t, 1) = 0 and initial data (u 10 , u 20 ) ∈ H × H with u 10 , u 20 ≥ 0. The functions f i,M , σi,M and h M here are given by f i,M (x, t, u) := f i (x, t, FM (u)), σi,M (x, t, u) := σi (x, t, FM (u)), h M (u, v) := h(FM (u), FM (v)). Proof. Note that existence and uniqueness for the problem on the infinite time interval [0, ∞) follow from existence and uniqueness on the finite time intervals [0, T ], for T > 0. Fix some T > 0 and some p > 8. We perform a Picard iteration in the space L p (Ω , C([0, T ]; H )) × L p (Ω , C([0, T ]; H )). Let v i,0 (t, x) = u i0 (x) for i = 1, 2. Given (v 1,n , v 2,n ) ∈ L p (Ω ; C([0, T ]; H )) × L p (Ω ; C([0, T ]; H )), we define z 1,n+1 and z 2,n+1 to solve the equations ∂z

) ∂(FM (v 1,n )) =∆z 1,n+1 − h M v 1,n (t, ·), v 2,n (t, ·) ∂x + f 1,M (x, t, v 1,n (t, ·)) + σ1,M (x, t, v 1,n (t, ·))W˙ ,

1,n+1

1 2 3 4 5 6

(

∂t

(5.1)

and

7

8

∂z

( ) ∂(FM (v )) =∆z 2,n+1 + h M v 1,n (t, ·), v 2,n (t, ·) ∂x + f 2,M (x, t, v 2,n (t, ·)) + σ2,M (x, t, v 2,n (t, ·))W˙ − , 2,n

2,n+1

∂t

(5.2)

with initial data u 10 and u 20 respectively. We then define, for i = 1, 2, wi,n+1 to be given by the solution to the obstacle problem i ∂wn+1

∂t ∫

T

0

∫

i i = ∆wn+1 + ηn+1 ,

10 11

i i wn+1 ≥ −z n+1 , 12

1

[

i wn+1 (t, x)

+

i z n+1 (t, x)

]

i ηn+1 (dx, dt)

= 0,

0

and set our (n + 1)th approximate solution to be given by (v 1,n+1 , v 2,n+1 ) := (z 1,n+1 + w1,n+1 , z 2,n+1 + w 2,n+1 ). For ease of notation, we define here ( ) gn (t, x) := h M v 1,n (t, ·), v 2,n (t, ·) FM (v 1,n (t, x)). Writing z

9

1,n+1

in mild form, we have that ∫ 1 z 1,n+1 (t, x) = G(t, x, y)u 10 (y)dy 0 ∫ t∫ 1 ∂G + (t − s, x, y)gn (s, y)dyds 0 0 ∂y ∫ t∫ 1 + G(t − s, x, y) f 1,M (y, s, v 1,n (s, ·))dxds 0 0 ∫ t∫ 1 + G(t − s, x, y)σ1,M (y, s, v 1,n (s, ·))W(dy, ds). 0

13 14

15

16

17

0

Note that, by Propositions 4.3, 4.4 and 4.5, we can see from this expression that z 1,n+1 ∈ C([0, T ]; H ) almost surely. Using the analogous expression for z 1,n and taking the difference, Please cite this article as: B. Hambly and J. Kalsi, Stefan problems for reflected SPDEs driven by space–time white noise, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.04.003.

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we see that [ ] p E ∥z 1,n+1 − z 1,n ∥H ,T [∫ ∫ p ]   t 1 ∂G  (t − s, x, y) [gn (s, y) − gn−1 (s, y)] dyds  ≤E   0 0 ∂y H ,T [∫ t ∫ 1  G(t − s, x, y) +E   0 0 p ]  [ ] 1,n 1,n−1 × f 1,M (y, s, v (s, ·)) − f 1,M (y, s, v (s, ·)) dxds  

(5.3)

H ,T

1

[∫ t ∫  +E  

[ ] G(t − s, x, y) σ1,M (y, s, v 1,n (s, ·)) − σ1,M (y, s, v 1,n−1 (s, ·)) 0 0 p ]  . × W(dy, ds)  H ,T

3

4

5

6

7

8

Note that ( ) ∥gn − gn−1 ∥H ,s ≤ C M,h ∥v 1,n − v 1,n−1 ∥H ,s + ∥v 2,n − v 2,n−1 ∥H ,s .

(5.4)

By making use of Propositions 4.3, 4.4 and 4.5, we are able to bound (5.3). We obtain that ∫ T [ ] [ p E ∥z 1,n+1 − z 1,n ∥H ,T ≤C T, p E ∥ f 1,M (·, s, v 1,n (s, ·)) 0 ] p ds − f 1,M (·, s, v 1,n−1 (s, ·)) ∥∞ ∫ T [ ] p E ∥gn − gn−1 ∥H ,s ds + 0 ∫ T [ E ∥ σ1,M (·, s, v 1,n (s, ·)) + 0 p ] −σ1,M (·, s, v 1,n−1 (s, ·)) ∥H ds. By the Lipschitz-type conditions on f, σ and the inequality (5.4), we see that this is at most ∫ T [ ] p p C M,h,T, p E ∥v 1,n − v 1,n−1 ∥H ,s + ∥v 2,n − v 2,n−1 ∥H ,s ds. 0

9 10

11

[ ] p By arguing in the same way, we obtain the same bound for E ∥z 2,n+1 − z 2,n ∥H ,T . Adding these together gives that [ ] [ ] p p E ∥z 1,n+1 − z 1,n ∥H ,T + E ∥z 2,n+1 − z 2,n ∥H ,T ∫ T (5.5) [ ] [ ] p p ≤ C M,h,T, p E ∥v 1,n − v 1,n−1 ∥H ,s + E ∥v 2,n − v 2,n−1 ∥H ,s ds. 0

12

13

14

15

By Theorem 3.2, we have that for i = 1, 2, ∥wi,n+1 − wi,n ∥H ,T ≤ ∥z i,n+1 − z i,n ∥H ,T almost surely. Therefore, for i = 1, 2, ∥v i,n+1 − v i,n ∥H ,T ≤ 2∥z i,n+1 − z i,n ∥H ,T

(5.6)

Please cite this article as: B. Hambly and J. Kalsi, Stefan problems for reflected SPDEs driven by space–time white noise, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.04.003.

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almost surely. It follows from (5.5) and (5.6) that [ ] [ ] p p E ∥v 1,n+1 − v 1,n ∥H ,T + E ∥v 2,n+1 − v 2,n ∥H ,T ∫ T [ ] [ ] p p ≤ C M,h,T, p E ∥v 1,n − v 1,n−1 ∥H ,s + E ∥v 2,n − v 2,n−1 ∥H ,s ds.

1

2

0

By iterating this inequality, we obtain that [ ] p p E ∥v 1,n+1 − v 1,n ∥H ,T + ∥v 2,n+1 − v 2,n ∥H ,T ∫ T [ ] p p ≤C M, p,T E ∥v 1,n − v 1,n−1 ∥H ,s + ∥v 2,n − v 2,n−1 ∥H ,s ds 0 ∫ T∫ s [ ] p p 2 ≤C M, p,T E ∥v 1,n−1 − v 1,n−2 ∥H ,u + ∥v 2,n−1 − v 2,n−2 ∥H ,u du ds 0 0 ∫ T∫ T [ ] p p 2 =C M, E ∥v 1,n−1 − v 1,n−2 ∥H ,u + ∥v 2,n−1 − v 2,n−2 ∥H ,u ds du p,T 0 u ∫ T [ ] p p 2 =C M, E ∥v 1,n−1 − v 1,n−2 ∥H ,u + ∥v 2,n−1 − v 2,n−2 ∥H ,u (T − u) du. p,T

3

4

0

Therefore, [ ] p p E ∥v 1,n+1 − v 1,n ∥H ,T + ∥v 2,n+1 − v 2,n ∥H ,T ∫ T [ ] (T − s)n−1 p p n ds ≤ C M, p,T E ∥v 1,1 − v 1,0 ∥H ,s + ∥v 2,1 − v 2,0 ∥H ,s (n − 1)! 0 [ 1,1 ] Tn p p n . ≤ C M, − v 1,0 ∥H ,s + ∥v 2,1 − v 2,0 ∥H ,s p,T × E ∥v n! Hence, for m > n ≥ 1, we have that [ ] p p E ∥v 1,n+1 − v 1,n ∥H ,T + ∥v 2,n+1 − v 2,n ∥H ,T [ k ] m−1 k ∑ C˜ M, [ ] p,T T p p ≤ E ∥v11 − v01 ∥H ,T + ∥v12 − v02 ∥H ,T → 0. k! k=n as m, n → ∞. Therefore, the sequence (v 1,n , v 2,n ) is Cauchy in the space L p (Ω ; C([0, T ]; H ))2 and so converges to some pair (v 1M , v 2M ). We now verify that this is indeed a solution to our evolution equation. Let (z 1M , z 2M ) solve the SPDEs ∂z 1M

∂(FM (v 1M ))

= ∆z 1M − p ′ (t) + f 1,M (x, t, v 1M ) + σ1,M (x, t, v 1M )W˙ , ∂t ∂x ∂z 2M ∂(FM (v 2M )) = ∆z 2M + p ′ (t) + f 2,M (x, t, v 2M ) + σ2,M (x, t, v 2M )W˙ − , ∂t ∂x where

5

6

7

8

9 10 11

12

13

14

′

p (t) :=

h M v 1M (t, ·), v 2M (t, ·) (

)

.

We then define (wiM , ηiM ) as solutions to the obstacle problem, with obstacles z iM , and set v˜ iM := z iM + wiM . Therefore, we have that ∂ v˜ 1M ∂(FM (v 1M )) = ∆v˜ 1M − p ′ (t) + f 1,M (x, t, v 1M ) + σ1,M (x, t, v 1M )W˙ + η1M , ∂t ∂x ∂ v˜ 2M ∂(FM (v 2M )) = ∆v˜ 2M + p ′ (t) + f 2,M (x, t, v 2M ) + σ2,M (x, t, v 2M )W˙ + η2M . ∂t ∂x Please cite this article as: B. Hambly and J. Kalsi, Stefan problems for reflected SPDEs driven by space–time white noise, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.04.003.

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By reproducing the same argument as that used to achieve the estimate (5.6), we are able to show that [ ] p p E ∥v˜ 1M − v 1,n ∥H ,T + ∥v˜ 2M − v 2,n ∥H ,T ∫ T [ ] p p ≤ C M,h,T, p E ∥v 1M − v 1,n ∥H ,s + ∥v 2M − v 2,n ∥H ,s ds → 0. 0

4 5 6 7 8

9

It follows that v˜ = v 1 in L p (Ω ; C([0, T ]; H )). The same applies to v 2 , so it follows that the pair (v 1 , v 2 ), together with the reflection measures (η1M , η2M ), do indeed satisfy our problem. Uniqueness follows by essentially the same argument. Given two solutions with the same initial data, (v11 , v12 ) and (v21 , v22 ) (together with their reflection measures), we argue as before to obtain that, for t ∈ [0, T ], ∫ t [ 1 [ ] ] p p 1 p 2 2 p E ∥v1 − v2 ∥H ,t + ∥v1 − v2 ∥H ,t ≤ E ∥v11 − v21 ∥H ,s + ∥v12 − v22 ∥H ,s ds. 1

0

10

11 12 13

14 15

16

The equivalence then follows by Gronwall’s inequality. □ Before stating and proving our main theorem, we first prove the following technical result which gives us a universal representation for terminal times of maximal solutions, and a canonical localising sequence. Proposition 5.4. Let (v 1 , η1 , v 2 , η2 ) be a maximal solution to the moving boundary problem in the relative frame. Then we have that the terminal time, τ , is given by τ := sup τ M , M>0

17

18

19

20 21 22 23 24 25 26 27 28 29 30 31

32

33 34 35

where { } τ M := inf t ≥ 0 | ∥v 1 ∥H ,t + ∥v 2 ∥H ,s ≥ M . Furthermore, τ M can be taken as a localising sequence for the solution. Proof. Let σn be a localising sequence for the solution. Define σ := supn≥1 σn . Fix some M > 0 and consider the localising sequence (σn ∧ τ M )n≥1 for the solution, with this solution on a potentially smaller stochastic interval. Taking the limit as n → ∞ for each M > 0, we obtain a local solution which agrees with (v 1 , η1 , v 2 , η2 ) until τ ∧ σ (its own terminal time), and has the localising sequence (σ ∧τ M ) M>0 . Arguing as in Theorem 5.3, for every M > 0, this must agree with the M-truncated problem until the time σ ∧ τ M . Therefore, for every M > 0, we have that σ ≥ τ M , as otherwise we could use the solution to the M-truncated problem to propagate the solution to a later time. Hence, the localising sequence σ ∧ τ M is simply just τ M . In addition, we have that σ ≥ τ M for every M almost surely, which implies that σ ≥ τ almost surely. Clearly, by agreement with the M-truncated problems until the times τ M , we have that τ ≥ σ , since the solution cannot be propagated beyond a blow-up in the H -norm. We therefore have the result. □ We are now in position to prove our main result, Theorem 2.11. Proof of Theorem 2.11. We prove existence of a maximal solution by concatenating solutions to the truncated problems. For M > 0, let V M := (v 1M , η1M , v 2M , η2M ) solve the M-truncated problem as in Theorem 5.3. Note that for M1 ≥ M2 > 0, V M1 solves (2.2) until the random Please cite this article as: B. Hambly and J. Kalsi, Stefan problems for reflected SPDEs driven by space–time white noise, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.04.003.

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stopping time

1

{

τ := inf t ∈ [0, ∞) | max

{

∥v 1M1 ∥H ,t , ∥v 2M1 ∥H ,t

}

}

≥ M2 .

Let z iM j solve the unreflected SPDEs associated with v iM j for i, j = 1, 2, as in Eq. (5.2). By considering the difference between z iM1 (t, x)1{t≤τ } and z iM2 (t, x)1{t≤τ } for i = 1, 2, we are able to argue as in the proof of Theorem 5.3 to obtain that, for T > 0, p > 8 and t ∈ [0, T ] [ ] p p E ∥z 1M1 − z 1M2 ∥H ,τ ∧t + ∥z 2M1 − z 2M2 ∥H ,τ ∧t ∫ t [ ] p p ≤ C T, p E ∥z 1M1 − z 1M2 ∥H ,τ ∧s + ∥z 2M1 − z 2M2 ∥H ,τ ∧s ds.

2

3 4 5

6

0

Gronwall’s lemma then applies to give that z iM1 = z iM2 almost surely for i = 1, 2, until the random time τ ∧ T . Since (v iM j , ηiM j ) are simply the solutions to the obstacle problem with obstacles z iM j , it follows that (v 1M1 , η1M1 , v 2M1 , η2M1 ) = (v 1M2 , η1M2 , v 2M2 , η2M2 ) until time τ ∧ T . This holds for every T > 0, so we have that they almost surely agree until time τ . We have therefore shown that the solutions to our M-truncated problems are consistent. This allows us to concatenate them. We define the process (v 1 , η1 , v 2 , η2 ) such that, for all M > 0, it agrees with (v 1M , η1M , v 2M , η2M ) until the time { { } } τ M := inf t ∈ [0, ∞) | max ∥v 1M ∥H ,t , ∥v 2M ∥H ,t ≥ M . Note that τM

7 8 9 10 11 12 13

14

15

{ { } } = inf t ∈ [0, ∞) | max ∥v 1 ∥H ,t , ∥v 2 ∥H ,t ≥ M .

16

Since V M solves (2.2) until time τ M , we have that (v , η , v , η ) solves (2.2) on the random interval [0, τ˜ ), where 1

1

2

2

τ˜ = sup τ M ,

17 18

19

M>0

with localising sequence τ M . This solution is maximal since, on the set {τ˜ < ∞}, we have that { } max ∥v 1 ∥H ,t , ∥v 2 ∥H ,t → ∞ as t ↑ τ˜ almost surely. Therefore, we know that a maximal solution exists. Uniqueness follows by the same arguments as those made for consistency among solutions to the truncated problems. Fix T > 0 and let (v˜ 1 , η˜ 1 , v˜ 2 , η˜ 2 ) be a maximal solution to (2.2). Note that the localising sequence can be taken as in Proposition 5.4. We then have that for every M > 0, it must agree with the solution to the M-truncated problem, and therefore with (v 1 , η1 , v 2 , η2 ), until { { } } inf t ∈ [0, T ] | max ∥v˜ 1 ∥H ,t , ∥v˜ 2 ∥H ,t ≥ M . Uniqueness then follows. □ We can now show that, as one might expect, blow up for the problem occurs precisely when the boundary speed blows up. Proposition 5.5. Let (v 1 , η1 , v 2 , η2 ) be a solution to the moving boundary problem in the relative frame on the maximal interval [0, τ ). Then τ = sup σ M , M>0 Please cite this article as: B. Hambly and J. Kalsi, Stefan problems for reflected SPDEs driven by space–time white noise, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.04.003.

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where { } σ M = inf t ∈ [0, ∞) | h(v 1 (t, ·), v 2 (t, ·)) ≥ M . Proof. By Proposition 5.4, and since h is bounded on bounded sets in H × H , it is sufficient to show that for every M > 0, σ M ≤ τ almost surely. Suppose that, for every T > 0, [ ] [ 1 ] p p E sup ∥v (t, ·)∥H + ∥v 2 (t, ·)∥H < ∞. (5.7) t∈[0,(σ M ∧τ ∧T ))

6

Then, since

7

sup

[ 1 ] ∥v (t, ·)∥H + ∥v 2 (t, ·)∥H = ∞

t∈[0,τ ∧T ) 8 9

10

11

12

13 14

15

almost surely on the set that {τ ≤ T } we must have that σ M < τ ∧ T almost surely on the set {τ ≤ T }. Therefore, for every T > 0, we would have that σ M 1{τ ≤T } < τ ∧ T almost surely. Letting T ↑ ∞ (via a countable sequence), we obtain that σ M 1{τ <∞} ≤ τ almost surely, so that σ M ≤ τ almost surely, giving the result. So it is sufficient to prove that (5.7) holds for every M, T > 0. Fix some T > 0 and some M > 0. By Theorem 3.2, [ ] [ ] E

sup t∈[0,(σ M ∧τ ∧T ))

16 17

18

19

20

21

22

p

∥v 1 (t, ·)∥H

24

sup t∈[0,(σ M ∧τ ∧T ))

p

∥z 1 (t, ·)∥H

,

where z 1 solves the unreflected problem corresponding to v 1 on the random interval [0, τ ∧ T ). That is, z 1 solves ∂v 1 ∂z 1 = ∆z 1 − h(v 1 (t, ·), v 2 (t, ·)) + f (x, t, v 1 (t, ·)) + σ (x, t, v 1 (t, ·))W˙ (dx, dt), ∂t ∂x on the interval [0, τ ∧ T ), where we take the localising sequence { { } } τ N := inf t ∈ [0, ∞) | max ∥v 1 ∥H ,t + ∥v 2 ∥H ,t ≥ N . Writing z 1 in mild form, we have that for t ∈ [0, T ] z 1 (t, x)1{t<σ M ∧τ N ∧T } = [∫ 1 ] 1{t<σ M ∧τ N ∧T } G(t, x, y)u 10 (y)dy + 0 [∫ t ∫ 1 ] ∂G 1{t<σ M ∧τ N ∧T } (t − s, x, y)h(v 1 (s, ·), v 2 (s, ·))v 1 (s, y)dyds + 0 0 ∂y [∫ t ∫ 1 ] 1{t<σ M ∧τ N ∧T } G(t − s, x, y) f (y, s, v 1 (s, ·))dxds + 0 0 [∫ t ∫ 1 ] 1{t<σ M ∧τ N ∧T } G(t − s, x, y)σ (y, s, v 1 (s, ·))W(dy, ds) , 0

23

≤ 2pE

(5.8)

0

We aim to bound the L p (Ω ; L ∞ ([0, T ]; H )) norms of the four terms on the right hand side uniformly over N . The first term is simple, so we omit the proof for this. For the second term, Please cite this article as: B. Hambly and J. Kalsi, Stefan problems for reflected SPDEs driven by space–time white noise, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.04.003.

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note that

1

] ∂G 1 2 1 1{t<σ M ∧τ N ∧T } (t − s, x, y)h(v (s, ·), v (s, ·))v (s, y)dyds 0 0 ∂y [∫ t ∫ 1 ] ∂G 1 2 1 ≤ (t − s, x, y)h(v (s, ·), v (s, ·))v (s, y)1{s<σ M ∧τ N ∧T } dyds . 0 0 ∂y By the definition of σ M , this is at most ] [∫ t ∫ 1 ∂G 1 (t − s, x, y)v (s, y)1{s<σ M ∧τ N ∧T } dyds . J (t, x) := C M 0 0 ∂y An application of Proposition 4.4 then gives that ] ∫ t [ [ ] p p 1 E ∥J ∥H ,t ≤ C M,T, p E sup ∥v (r, ·)1{r <σ M ∧τ N ∧T } ∥H ds. [∫ t ∫

1

3

4

5

6

r ∈[0,s]

0

Arguing in the same way, applying Proposition 4.3 in place of Proposition 4.4, we are able to control the third term of (5.8). For the fourth term of (5.8), we note that ⏐∫ t ∫ 1 ⏐ ⏐ ⏐ 1 ⏐ 1{t<σ M ∧τ N ∧T } ⏐ G(t − s, x, y)σ (y, s, v (s, ·))W(dy, ds)⏐⏐ 0 ⏐ ⏐∫ 0t ∫ 1 ⏐ ⏐ 1 ⏐ G(t − s, x, y)σ (y, s, v (s, ·))1{s<σ M ∧τ N ∧T } W(dy, ds)⏐⏐ ≤⏐ 0

2

7 8

9

0

=: K (t, x). Applying Proposition 4.5 and noting the linear growth condition on σ then gives that, for t ∈ [0, T ] [ ] ] ∫ t [ [ ] p p 1 E ∥K ∥H ,t ≤ C T, p 1 + E sup ∥v (r, ·)1{r <σ M ∧τ N ∧T } ∥H ds . r ∈[0,s]

0

Putting the four terms together, we have that, for t ∈ [0, T ], [ ]  1 p E sup z (r, ·)1{r <σ ∧τ ∧T }  M

r ∈[0,t]

[

∫

t

≤ C T, p,M 1 +

H

N

[

] sup ∥v

E

1

r ∈[0,s]

0

Therefore, [

p (r, ·)1{r <σ M ∧τ N ∧T } ∥H

] ds .

]  p sup v 1 (r, ·)1{r <σ M ∧τ N ∧T } H

E

r ∈[0,t]

[

∫

≤ C T, p,M 1 +

[

t

E 0

] sup

r ∈[0,s]

p ∥v 1 (r, ·)1{r <σ M ∧τ N ∧T } ∥H

] ds .

It follows by Gronwall’s inequality that [ ] p  1   sup E sup v (t, ·) N >0

t∈[0,σ M ∧τ N ∧T )

H

[ = sup E N >0

]  p sup v 1 (t, ·)1{t<σ M ∧τ N ∧T } H

≤ K T, p,M < ∞.

t∈[0,T ]

Please cite this article as: B. Hambly and J. Kalsi, Stefan problems for reflected SPDEs driven by space–time white noise, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.04.003.

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Note that, by the MCT [ sup E

N >0 3

4 5 6

7

8 9

10 11

12 13

14

sup t∈[0,σ M ∧τ N ∧T )

]  1  v (t, ·) p H

]

[ =E

sup t∈[0,σ M ∧τ ∧T )

 1  v (t, ·) p H

The same estimates hold for v 2 . This concludes the proof.

□

Remark 5.6. The previous result gives that solutions to the moving boundary problem can only blow-up in ways which cause the boundary speed to blow up. For example, if we have the classical boundary behaviour given by h(u, v) = γ (u ′ (0) − v ′ (0)), for a constant, γ , we know that blow-up can only occur when one of the spatial derivatives at the shared interface blow up. Proposition 5.7. Suppose that the function h is globally bounded. Then the unique maximal solution to (2.2) is global i.e. τ = ∞ almost surely. Proof. The proof is essentially the same as the proof of Proposition 5.5. By following the same steps as in that proof, we are able to show that, for every T > 0 [ ] [ 1 ] p p E sup ∥v (t, ·)∥H + ∥v 2 (t, ·)∥H < ∞. (5.9) t∈[0,τ ∧T )

18

The key difference here is that in Proposition 5.5, the non-Lipschitz term from the moving boundary was controlled by noting that h(v 1 (t, ·), v 2 (t, ·)) is bounded on the time intervals [0, σ M ). We need not concern ourselves with that detail here, as we assume that h is globally bounded from the outset. It follows from (5.9) that the solution must be global. □

19

6. Numerical illustrations

15 16 17

20 21 22 23 24 25 26 27 28 29 30 31

32

33 34 35 36

In this section, we implement a numerical scheme in order to illustrate some typical profiles for solutions to our Stefan problems and point out some of the features which were highlighted in the previous analysis. In particular, we are able to see the presence of spatial derivatives at the shared boundary when we choose the drift and volatility parameters appropriately. We will also √ contrast this with the behaviour at the boundary when the volatility only decays there like x, providing numerical evidence that a spatial derivative does not exist at the boundary in this case. This is of interest as we have not shown the linear decay condition which we imposed on the volatility to be optimal. To begin, we briefly describe the numerical scheme used. We will consider drift and volatility functions which only depend on the position in space (in the relative frame) and the value of the solution at that particular position at that particular time. We also choose these functions to be the same for both sides of the Stefan problem. In addition, our functions h which determine the boundary behaviour will take the classical form ( ) h(u, v) = γ u ′ (0) − v ′ (0) , where γ is a positive constant. We simulate the process on the interval [0, T ] and space interval [ p(t) − 1, p(t) + 1] at a given t ∈ [0, T ]. That is, we simulate the equation in the relative frame on [0, T ] × [−1, 1], with v 1 , v 2 supported on [0, T ] × [0, 1]. We discretise the equation into M time steps and N space steps, and define ti := i T /M, xi := i/N . Given our simulated solution Please cite this article as: B. Hambly and J. Kalsi, Stefan problems for reflected SPDEs driven by space–time white noise, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.04.003.

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Fig. 1. Static Snapshot of v 1 at time t = 0.1 on [0, 1] and on [0, 0.04], in the case where σ = σa (x), so that σ decays linearly at the boundary.

up to the jth time step, we define the values of the approximate solution at the ( j + 1)th time step by setting ⏐ ⏐ ) N2 ( 1 1 v (t j+1 , xi ) := ⏐⏐v 1 (t j , xi ) + v (t j , xi+1 ) + v 1 (t j , xi−1 ) − 2v 1 (t j , xi ) M )( ) N2 ( 1 −γ v (t j , x1 ) − v 2 (t j , x1 ) v 1 (t j , xi+1 ) − v(t j , xi ) M ⏐ √ ⏐ T N 1 ⏐ + f (xi , v 1 (t j , xi )) + √ σ (xi , v 1 (t j , xi ))Z i, j ⏐ , ⏐ M M where the Z i, j are independent unit normal random variables. v 2 (t j+1 , xi ) is defined similarly. We note that taking the absolute value on the right hand side here is intended to capture the fact that the process is reflected at zero. Given our simulated processes v 1 and v 2 , we are then able to reproduce the boundary process by noting that p ′ (t) = h(v 1 (t, ·), v 2 (t, ·)). We can therefore take

1 2

3

4 5 6 7 8

j

p(t j ) := γ

∑ N ( ) v 1 (tk , x1 ) − v 2 (tk , x1 ) . M k=1

As we are interested in the effects of different volatility functions on our equations, we will fix the other parameters here. Throughout all of our simulations, we will choose the drift function f and the parameter γ such that f = 1 and γ = 10. The initial data for both v 1 and v 2 will be given by the function u 0 , where u 0 (0) = u 0 (1) = 0, and u 0 (0.5) = 1, with linear interpolation in between these points. The volatility functions for which we perform our simulations are given by σa (x), σb (u) and σc (x), where 1. σa (x) is linear between the points 0, 0.5 and 1, with σa (0) = 0, σa (0.5) = σa (1) = 1. 2. σb (u) = √ u. 3. σc (x) = x. The following figures depict the results of our simulations. We see in Fig. 1 that the profile is less volatile as the boundary x = 0 is approached, and the derivative at the boundary is visible. The boundary function appears to be smooth to some Please cite this article as: B. Hambly and J. Kalsi, Stefan problems for reflected SPDEs driven by space–time white noise, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.04.003.

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Fig. 2. 3D plot of v 1 and plot of the boundary motion, p(t), in the case where σ = σa (x), so that σ decays linearly at the boundary.

Fig. 3. Static Snapshot of v 1 at time t = 0.1 on [0, 1] and on [0, 0.04], in the case where σ = σb (u) = u.

Fig. 4. 3D plot of v 1 and plot of the boundary motion, p(t), in the case where σ = σb (u) = u.

1 2 3

degree in Fig. 2(b), as expected. Our second set of simulations deals with the case when the volatility is multiplicative, which also falls within the framework of our earlier analysis. Figs. 3 and 4 show the results of our simulation in this case. Please cite this article as: B. Hambly and J. Kalsi, Stefan problems for reflected SPDEs driven by space–time white noise, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.04.003.

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Fig. 5. Static Snapshot of v 1 at time t = 0.1 on [0, 1] and on [0, 0.04], in the case where σ = σc (x) =

Fig. 6. 3D plot of v 1 and plot of the boundary motion in the case where σ = σc (x) =

√

√

x.

x.

The presence of spatial derivatives at both ends of the profile is visible in Fig. 3(a). This is simply because the equation is symmetric. The Dirichlet condition is imposed at both 0 and 1, so that the volatility decays sufficiently quickly at both ends.√ We now present the case where the volatility decays like x at the shared boundary, which falls outside of our proof for existence and uniqueness (see Figs. 5 and 6). In this case, the profiles can be seen to be significantly rougher close to the boundary. They do not appear to have spatial derivatives there (although this is inconclusive of course). Interestingly, the boundary motion appears to be rough to some degree. We note, however, that the scheme did not blow up — we were able to obtain sensible profiles and sensible processes for the movement of the boundary. This suggests that, perhaps by considering the derivative at the boundary in a suitable weak sense, equations of this form may be solvable without the need for linear decay of the volatility, and could produce rough paths for the boundary motion. We perform a simple numerical test in order to give some (albeit naive) quantification of how linear the profiles are close to the boundary in the cases presented above. For a profile at a given time, we fit the values in the spatial interval [0, 0.04] to a line passing through the origin by the method of least squares. Calculating the sum of the squared residuals and dividing by the square of height of the fitted line at position 0.04, we obtain values indicating how well the profile can be fitted to a line, with these values now independent of scale and so comparable Please cite this article as: B. Hambly and J. Kalsi, Stefan problems for reflected SPDEs driven by space–time white noise, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.04.003.

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B. Hambly and J. Kalsi / Stochastic Processes and their Applications xxx (xxxx) xxx Table 1 Time-averaged measure of deviation of profile from linear fit on the spatial interval [0, 0.04]. σa (x)

σb (u)

σc (x)

0.04274

0.01473

0.28378

3

across different profiles. Time averaging these values for each of the three cases, we obtain the following values, which clearly √ indicate that there is significant deviation from being linear close to zero when σ = σc (x) = x (see Table 1).

4

Acknowledgement

1 2

5

6

The research of J. Kalsi was supported by EPSRC (EP/L015811/1). Appendix. Proof of Proposition 4.2

15

We provide the main details for the proofs of the inequalities stated in Proposition 4.2. Recalling Remark 4.1, we focus on proving the estimates for the component J of G, where J is given by [ ( ) ( )] 1 (x − y)2 (x + y)2 J (t, x, y) = √ exp − − exp − . 4t 4t 4π t We also define J˜ such that y J˜(t, x, y) := J (t, x, y). (A.1) x The proofs of the estimates in Lemma 3.2 of [10] were very helpful for producing the following arguments.

16

A.1. Proof of inequality 1

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Proof. The first inequality in the Proposition states that, for every T > 0, there exists a constant C T such that, for every t ∈ (0, T ], ∫ 1 CT 1 G(t, x, y)dy ≤ √ . sup x t x∈(0,1] 0 We prove the corresponding bound for J . [ ( ) ( )] 1 1 J (t, x, y) = √ exp −(x − y)2 /4t − exp −(x + y)2 /4t x x 4π t ( )[ ] 1 = √ exp −(x − y)2 /4t 1 − exp (−x y/t) . x 4π t Note that, since x ≥ 0, x y = x((y − x) + x) ≤ 2x 2 1{y−x≤x} + 2x(y − x)1{y−x>x} . Therefore ( ) (1 − exp(−x y/t)) ≤ 1 − exp(−2x 2 /t) 1{y−x≤x} + (1 − exp(−2x(y − x)/t)) 1{y−x>x} . Please cite this article as: B. Hambly and J. Kalsi, Stefan problems for reflected SPDEs driven by space–time white noise, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.04.003.

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Since, for x ≥ 0 (1 − e

−x

1

√ ) ≤ min(x,

x),

2

we have that

3

] x|y − x| x . (1 − exp(−x y/t)) ≤ 2 √ + t t Putting these bounds together, we see that ( )[ ] ∫ 1 ∫ 1 1 1 (x − y)2 |y − x| 1 J (t, x, y) dy ≤ C exp − + dy √ √ 4t t t t 0 x 0 ] ∫ [ 1 −y 2 /4t |y| C 2 ≤C e + 3/2 e−y /4t dy = √ . □ t t R t [

4

5

6

A.2. Proof of inequality 3

7

Proof. We begin by recalling the inequality. For every T > 0 and q ∈ (1, 2), we claim that ∃C T,q such that ∫ t [∫ 1 ( )2 ]q ˜ x, z) − G(s, ˜ sup G(s, y, z) dz ds ≤ C T,q |x − y|(2−q)/3 . t∈[0,T ] 0

8 9

10

0

We will prove the corresponding bound for J˜. The arguments made in the proof of Lemma 3.2.1 of [10] are followed here, and we make adjustments where necessary. Assume wlog that x ≤ y, and define h = y − x. By following the proofs of inequalities (1) and (2) in Lemma 3.2.1 of [10], we arrive at the inequality ) [ ( ∫ 1( )2 −(x+h)2 /4s −x 2 /4s C 1 − e 1 − e 0≤ J˜(s, x, z) − J˜(s, y, z) dz ≤ √ 1 + s + x2 (x + h)2 s 0 ( x(x+h) )] 1 − e− 4s −h 2 /16s 1 + 2s −e x(x + h)

11 12 13 14

15

=:C R(s, x, h). Therefore, it is enough to prove that ∫ T R(s, x, h)q ds ≤ C T,q h (2−q)/3 .

16

17

0

In the spirit of [10] once again, we control this integral by proving two separate estimates for it which we will then combine. These estimates are obtained by splitting R up into different components. Note that R(s, x, h) = R1 (s, x, h) + R2 (s, x, h) ≥ 0,

18 19 20

21

where we define

22

) 1 ( 2 R1 (s, x, h) := √ 1 − e−h /16s , s

23

and

24

[

R2 (s, x, h) :=

−x 2 /4s

√ 1−e s x2

−(x+h)2 /4s

+

1−e (x + h)2

( − 2e−h

2 /16s

− x(x+h) 4s

1−e x(x + h)

)] .

Please cite this article as: B. Hambly and J. Kalsi, Stefan problems for reflected SPDEs driven by space–time white noise, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.04.003.

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Therefore, if I (s, x, h) ≥ 0 such that R2 (s, x, h) ≤ I (s, x, h) we have that ( ) R(s, x, h)q ≤ (R1 (s, x, h) + I (s, x, h))q ≤ Cq R1 (s, x, h)q + I (s, x, h)q . Our approach will therefore be to bound the integral of the R1 and I terms separately, and we do so for two different functions I . Starting with the R1 term, we first note that 1 R1 (s, x, h) ≤ √ , s

and that R1 (s, x, h) ≤

h2 , 16s 3/2

where we once again make use of the inequality (1 − e−x ) ≤ x. It then follows that ) ∫ ∞ ∫ T ∫ h2 ( h 2q 1 q ds, ds + R1 (s, x, h)q ds ≤ √ q 3q/2 s h 2 16 s 0 0 Calculating, we see that this is equal to Ch 2−q . For the R2 term, we have the following bound corresponding to our first choice of I , [ )] ( √ 1 x(x + h) 1 2 R2 (s, x, h) ≤ s − 2e−h /16s − , 2s 4s 32s 2 where we have made use of the fact that x2 ≤ 1 − e−x ≤ x. 2 In this case, I is then equal to ] x(x + h) 1 [ 2 2 e−h /16s . √ 1 − e−h /16s + 16s 3/2 2 s x−

The first of these terms is in the same form as R1 , and so we can control it in the same way. For the second term, we have that )q ( ( ) 2 ( s )q x(x + h) −h 2 /16s q e−(h )/16s q q q q e ≤ C x (x + h) = C x (x + h) q q 16s 3/2 s 3/2 s 3/2 h 2 = Cq x q (x + h)q s −q/2 h −2q ,

18

since e−x ≤ ∫ T( 0

19

1 . ex

This bound then gives that ) ) ∫ h2 ( x(x + h) −h 2 /16s q x(x + h) −h 2 /16s q e ds ≤ e ds 16s 3/2 16s 3/2 0 ) ∫ ∞( x(x + h) −h 2 /16s q + e ds 16s 3/2 h2 (∫ 2 ∫ h q q −q/2 −2q ≤ Cq x (x + h) s h ds + 0

20

21

22

23

∞ h2

1 s 3q/2

) ds .

Calculating gives that the right hand side is equal to Cq x q (x + h)q h 2−3q . So we obtain our first bound ∫ T [ ] R(s, x, h)q ds ≤ Cq h 2−q + x q (x + h)q h 2−3q .

(A.2)

0 Please cite this article as: B. Hambly and J. Kalsi, Stefan problems for reflected SPDEs driven by space–time white noise, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.04.003.

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We now prove our second estimate for the integral by bounding R2 with a different choice of I . We have that ⏐ ( x(x+h) )⏐ 2 2 ⏐ √ ⏐⏐ 1 − e−x /4s 1 − e−(x+h) /4s 1 − e− 4s ⏐ + − 2 R2 (s, x, h) ≤ s ⏐ ⏐ ⏐ x2 (x + h)2 x(x + h) ⏐ ⏐ (A.3) ( )⏐ ⏐ ) 1 − e− x(x+h) 4s √ ⏐⏐ ( 2 ⏐ + s ⏐2 1 − e−h /16s ⏐. ⏐ x(x + h) ⏐ The second term on the right hand side is at most ) C ( 2 √ 1 − e−h /16s . s

3

5

1 − e−x/4t . x

Then we can calculate that, for x ∈ (0, 1] and every s > 0, ⏐ ⏐ ⏐ ⏐ ⏐ ∂φ ⏐ ⏐ ⏐ ⏐ (s, x)⏐ = ⏐− 1 (1 − e−x/4s ) + 1 e−x/4s ⏐ ≤ C . ⏐ ∂x ⏐ ⏐ x2 ⏐ xs 4xs

2

4

This is once again in the form of R1 , and so can be controlled in the same way. In order to bound the first term on the right hand side of (A.3), we define here the function φ(t, x) :=

1

6 7

8

9

(A.4)

Therefore, ⏐ ( x(x+h) )⏐ 2 2 ⏐ √ ⏐⏐ 1 − e−x /4s 1 − e−(x+h) /4s 1 − e− 4s ⏐ s⏐ + − 2 ⏐ 2 2 ⏐ x (x + h) x(x + h) ⏐ ⏐ √ ⏐ = s ⏐φ(s, x 2 ) + φ(s, (x + h)2 ) − 2φ(s, x(x + h))⏐ . Using (A.4), we see that this is at most ( ) C h h2 2 × (xh + h ) = C + . √ √ √ x2 s x s x2 s We have that )]q ( q ) ∫ T [( h2 h h h 2q + ds = C C + . √ √ T,q xq x 2q x s x2 s 0 Putting these parts together, we have deduced our second estimate, ( ) ∫ T hq h 2q R(s, x, h)q ds ≤ C T,q h 2−q + q + 2q . x x 0 We can now conclude our proof by splitting into cases, and using the two inequalities in the 4q−2 different scenarios. For x ≤ h 3q , we use our first bound, (A.2), and obtain that ∫ T [ ] R(s, x, h)q ds ≤ Cq h 2−q + x q (x + h)q h 2−3q 0 [ ] 8q−4+6−9q 2−q 3 ≤ Cq h 2−q + h ≤ Cq h 3 , Please cite this article as: B. Hambly and J. Kalsi, Stefan problems for reflected SPDEs driven by space–time white noise, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.04.003.

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where we have used that, since q ∈ (1, 2), we have (4q−2)/3q ∈ (2/3, 1) and so h ≤ h (4q−2)/3q . 4q−2 If x > h 3q , we have by our second bound that ) ( ∫ T hq h 2q R(s, x, h)q ds ≤ C T,q h 2−q + q + 2q x x 0 ( ) 8q−4 2−q q− 4q−2 3 ≤ C T,q h +h + h 2q− 3 ( ) = C T,q h 2−q + h (2−q)/3 + h 2(2−q)/3 ≤ C T,q h (2−q)/3 ,

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recalling that h ∈ [0, 1] and q ∈ (1, 2). This concludes the proof.

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A.3. Proof of inequality 4

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Proof. We begin by recalling the inequality. For T > 0, 0 ≤ s ≤ t ≤ T and q ∈ (1, 2), ∃ C T,q > 0 such that ]q ∫ s [∫ 1 2 ˜ ˜ sup (G(t − r, x, y) − G(s − r, x, y)) dy dr ≤ C T,q |t − s|(2−q)/2 . x∈[0,1] 0

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Once again, we will prove the corresponding bound for J˜, given by (A.1). The proof will have two steps. We will first perform a change of variables, which will allow us to bound the integral by |t − s|(2−q)/2 multiplied by some integral which does not depend on t, s, which we denote by I (x). We will then show that sup I (x) < ∞. x≥0

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Let k := t − s. We have that ]q ∫ s [∫ 1 2 ˜ ˜ ( J (s − r + k, x, y) − J (s − r, x, y)) dy dr 0 0 ]q ∫ s [∫ 1 2 ˜ ˜ = ( J (r + k, x, y) − J (r, x, y)) dy dr 0 0 [ ) ( ))2 ]q ∫ s ∫ 1 ( ( x y r x y 1 ˜ r = J + 1, √ , √ − J˜ ,√ ,√ dy dr. k k k k k k 0 0 k √ Performing the change of variables z := y/ k and u = r/k, we obtain that this is at most k (2−q)/2 × I ( √xk ), where ∫ ∞ [∫ ∞ ( )2 ]q ˜ ˜ I (x) = J (u + 1, x, z) − J (u, x, z) dz du. 0

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By inequality (1) of Lemma 3.2.1 in [10], we have that there exists C > 0 such that for u ≥ 0 ∫ ∞ C sup J˜(u, x, z)2 dz ≤ √ . u x≥0 0 Therefore, for u ≥ 0 we have ∫ ∞( )2 C sup J˜(u + 1, x, z) − J˜(u, x, z) dz ≤ √ . u x≥0 0 Please cite this article as: B. Hambly and J. Kalsi, Stefan problems for reflected SPDEs driven by space–time white noise, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.04.003.

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It follows that, for q ∈ (1, 2), ∫ 1 [∫ ∞ ( )2 ]q ˜ ˜ J (u + 1, x, z) − J (u, x, z) dz du = Cq < ∞. 0

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So it is only left to control ∫ ∞ [∫ ∞ ( )2 ]q J˜(u + 1, x, z) − J˜(u, x, z) dz du. 1

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(A.5)

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We split the space integral here into two cases, when {z ≥ 2x}, and when {z < 2x}. For the integral over the spatial region {z < 2x}, we have that, ∫ ∞ [∫ 2x ( )2 ]q ˜ ˜ J (u + 1, x, z) − J (u, x, z) dz du 1 0 ]q ∫ ∞ [∫ ∞ 2 ≤Cq (J (u + 1, x, z) − J (u, x, z)) dz du ∫1 ∞ [∫0 ∞ [ ]2 ]q 1 −(x−z)2 /4(u+1) 2 e − e−(x−z) /4u dz du ≤Cq u 1 ∫ ∞0[∫ ∞ [ ]2 ]q 1 −(x+z)2 /4(u+1) 2 + Cq e − e−(x+z) /4u dz du u 1 0 ]q )2 ( ∫ ∞ [∫ ∞ ( )2 1 1 −(x−z)2 /4(u+1) −(x+z)2 /4(u+1) dz du + Cq e −e √ −√ u u+1 1 0 ∫ ∞ [∫ )2 ]q 1 ( −z 2 /4(u+1) −z 2 /4u ≤Cq e −e dz du u 1 [∫ ∞ ( ∫ ∞R )2 ]q 1 −(x−z)2 /4(u+1) −(x+z)2 /4(u+1) e − e dz du + Cq u 3q 0 1 ]q ∫ ∞ [∫ ∫ ∞ ) 1 − 1 ) 2 1 −z 2 /2(u+1) ( 1 −z 2 ( 4u 4(u+1) ≤Cq dz du + Cq du e 1−e 5q/2 u R u 1 1 ]q ∫ ∞ [∫ 4 ∫ ∞ ∫ ∞ z −z 2 /2(u+1) 1 1 ≤Cq e dz du + C du = C du. q q 5 5q/2 5q/2 u u u 1 R 1 1 This last term is finite and does not depend on x, so we have successfully bounded (A.5) in the case when {z < 2x}. On the set {z ≥ 2x}, we have that ∫ ∞ [∫ ∞ ( )2 ]q J˜(u + 1, x, z) − J˜(u, x, z) dz du 1 2x ∫ ∞ [∫ ∞ 2 (( z 1 2 ≤ Cq e−(x−z) /4(u+1) 2 u x 0 2x ) ( ))2 ]q 2 2 −(x+z)2 /4(u+1) −e − e−(x−z) /4u − e−(x+z) /4u dz du )2 ∫ ∞ [∫ ∞ 2 ( z 1 1 + Cq √ −√ 2 u u+1 2x x 0 ]q ( )2 −(x−z)2 /4(u+1) −(x+z)2 /4(u+1) × e −e dz du

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= Cq

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[∫

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]q ) z 2 1 ( −x z/(u+1) −x z/u 2 −(x−z)2 /2(u+1) du e − e e dz 2 1 2x x u ∫ ∞ [∫ ∞ 2 )2 ]q ) ( −(x−z)2 /4(u+1) z 1( −x z/u 2 −(x−z)2 /4u + Cq 1 − e e − e dz du 2 1 2x x u ]q ∫ ∞ [∫ ∞ 2 ) z 1 −(x−z)2 /2(u+1) ( −x z/(u+1) 2 + Cq e 1 − e dz du 2 3 1 2x x u

∫ ≤ Cq

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) z 2 1 ( −(x−z)2 /4(u+1) ( e 1 − e−x z/(u+1) 2 0 2x x u ]q ( )) 2 −(x−z)2 /4u −x z/u −e 1−e dz du [ )2 ∫ ∞ ∫ ∞ 2( 1 1 z + Cq − √ √ 2 u u+1 0 2x x ]q ( )2 −(x−z)2 /4(u+1) −(x+z)2 /4(u+1) dz du × e −e

∫

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[∫

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This is at most ]q ∫ ∞ [∫ ∞ 2 ) z 1 −2x z/(u+1) ( −x z/u(u+1) 2 −(x−z)2 /2(u+1) Cq e 1−e e dz du 2 2x x u 1 ∫ ∞ [∫ ∞ 4 )2 ]q z −(x−z)2 /2(u+1) ( −(x−z)2 /4u(u+1) e 1−e dz du + Cq 3 1 2x u ] ∫ ∞ [∫ ∞ q z4 2 e−(z−x) /2(u+1) dz du + Cq 3 (u + 1)2 u 1 2x ]q ∫ ∞ [∫ ∞ z4 −z 2 /8(u+1) ≤Cq e dz du u 3 (u + 1)2 1 0 ]q ∫ ∞ [∫ ∞ z8 −z 2 /8(u+1) + Cq e dz du u 5 (u + 1)2 1 0 ]q ∫ ∞ [∫ ∞ z4 −z 2 /8(u+1) + Cq e dz du u 3 (u + 1)2 0 ∫ ∞1 1 ≤Cq du = Cq < ∞. 5q/2 u 1 Therefore we have that supx≥0 I (x) < ∞, concluding the proof. □ A.4. Proof of inequality 5 Proof. We want to prove that, for t ∈ (0, T ] ∫ 1⏐ ⏐ CT ⏐ ⏐˜ sup ⏐ H (t, x, y)⏐ dy ≤ √ . t x∈[0,1] 0 We prove the corresponding bound for K , where we define, for x ∈ (0, 1], y ∈ [0, 1] and t ≥ 0, y ∂J K (t, x, y) := (t, x, y), x ∂y Please cite this article as: B. Hambly and J. Kalsi, Stefan problems for reflected SPDEs driven by space–time white noise, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.04.003.

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and for y ∈ [0, 1], t ≥ 0 we define ∂ J (t, 0, y). ∂ x∂t Note that, for t > 0, x ∈ (0, 1], y ∈ [0, 1] and y ≥ 2x, ] ] C y2 [ Cy [ 2 2 2 2 |K (t, x, y)| ≤ 3/2 e−(x−y) /4t + e−(x+y) /4t + 3/2 e−(x−y) /4t − e−(x+y) /4t t t x ] C y −y 2 /16t C y 2 −(x−y)2 /4t [ ≤ 3/2 e + 3/2 e 1 − e−x y/t t t x ] [ C y3 −y 2 /16t C y + 5/2 . ≤e t 3/2 t

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By letting x ↓ 0, we also have that the bound holds at x = 0 for t > 0 and y ∈ [0, 1]. For x ∈ (0, 1], y ∈ [0, 1] and y < 2x, we have that ⏐ ⏐ ⏐∂ J ⏐ |K (t, x, y)| ≤2 ⏐⏐ (t, x, y)⏐⏐ ∂y ] C [ 2 2 ≤ 3/2 (x − y)e−(x−y) /4t + (x + y)e−(x+y) /4t t ] C [ 2 2 ≤ 3/2 (x − y)e−(x−y) /16t + (x + y)e−(x+y) /16t t It follows that [ ] ∫ ∫ 1 y y3 C 2 |K (t, x, y)| dy ≤ C sup e−y /16t 3/2 + 5/2 dy ≤ √ . □ t t t x∈[0,1] 0 R References [1] T. Arnarson, J. Eriksson, On the size of the non-coincidence set of parabolic obstacle problems with applications to American option pricing, Math. Scand. 101 (1) (2007) 148. [2] R.C. Dalang, D. Khoshnevisan, E. Nualart, Hitting probabilities for systems of non-linear stochastic heat equations with additive noise, ALEA Lat. Am. J. Probab. Math. Stat. 3 (2007) 231–271. [3] C. Donati-Martin, E. Pardoux, White noise driven SPDEs with reflection, Probab. Theory Related Fields 95 (1993) 1–24. [4] Ben Hambly, Jasdeep Kalsi, A Reflected Moving Boundary Problem Driven by Space-Time White Noise, ar Xiv:1805.10166. [5] Martin Keller-Ressel, Marvin S. Müller, Stefan-type stochastic moving boundary problem, Stoch. Partial Differ. Equ. Anal. Comput. 4 (2015) 746–790. [6] Kunwoo Kim, Zhi Zheng, Richard B. Sowers, A stochastic Stefan problem, J. Theor. Probab. 25 (4) (2012) 1040–1080. [7] Marvin S. Müller, A stochastic Stefan-type problem under first order boundary conditions, Ann. Appl. Probab. (2019) forthcoming. [8] D. Nualart, E. Pardoux, White noise driven quasilinear SPDEs with reflection, Probab. Theory Related Fields 93 (1992) 77–89. [9] T. Xu, T. Zhang, White noise driven SPDEs with reflection: Existence, uniqueness and large deviation principles, Stoch. Process. Appl. 119 (10) (2009) 3453–3470. [10] Z. Zheng, Stochastic Stefan Problems: Existence, Uniqueness and Modeling of Market Limit Orders (Ph.D. thesis), Graduate College of the University of Illinois at Urbana-Champaign, 2012.

Please cite this article as: B. Hambly and J. Kalsi, Stefan problems for reflected SPDEs driven by space–time white noise, Stochastic Processes and their Applications (2019), https://doi.org/10.1016/j.spa.2019.04.003.

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